Originally published in: École d’Été de Probabilités de Saint-Flour XV-XVII –1985-87, Lecture Notes in Mathematics, Vol. 1362, 277–425, DOI: 10.1007/ BFb0086183, © Springer-Verlag Berlin Heidelberg 1988, Reprint by Springer-Verlag Berlin Heidelberg 2013

113

GEOMETRIC ASPECTS OF DIFFUSIONS ON MANIFOLDS

David ELWORTHY

114

D. ELWORTHY : "GEOMETRIC ASPECfS OF DIFFUSIONS ON MANIFOLDS"

Introduction

CHAPTER I: STOCHASTIC DIFFERENTIAL EQUATIONS AND MANIFOLDS

§ 1. Some notation, running hypotheses, and basic facts about manifolds. 282

§2. Stochastic differential equations on M . 285

§3. An Ito formula. 288

§4. Solution flows. 290

CHAPTER II : SOME DIFFERENTIAL GEOMETRY FOR PRINCIPAL

BUNDLES AND CONSTRUCTIONS OF BROWNIAN MOTION

§ 1. Connections on principal bundles and covariant differentiation. 296

§2. Horizontal lifts, covariant derivatives, geodesics, and a second form of the Ito formula. 299

§3. Riemannian metrics and the Laplace-Beltrami operator. 304

§4. Brownian motion on M and the stochastic development. 307

§5. Examples: Spheres and hyperbolic spaces. 312

§6. Left invariant SDS on Lie groups.

§7. The second fundamental form and gradient SDS for an embedded submanifold.

§8. Curvature and the derivative flow.

316

316

319

§9. Curvature and torsion forms. 323

§10. The derivative of the canonical flow. 327

CHAPTER III : CHARACTERISTIC EXPONENTS FOR STOCHASTIC FLOWS

§1. The Lyapunov spectrum. 331

§2. Mean exponents. 335

§3. Exponents for gradient Brownian flows: the difficulties of estimating exponents in general. 339

§4. Exponents for canonical flows. 342

§5. Moment exponents. 356

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CHAPTER IV : THE HEAT FLOW FOR DIFFERENTIAL FORMS AND THE

TOPOLOGY OF M

§ 1. A class of semigroups and their solutions.

§2. The top of the spectrum of ~.

§3. Bochner theorems for L2 harmonic forms.

§4. de Rham cohomology, Hodge theory and cohomology with compact support.

§5. Brownian motion and the components of M at infinity.

CHAPTER V: HEAT KERNELS: ELEMENTARY FORMULAE,

INEQUALITIES, AND SHORT TIME BEHAVIOUR

§ 1. The elementary formula for the heat kernel for functions.

§2. General remarks about the elementary formula method and its extensions.

§3. The fermionic calculus for differential forms, and the Weitzenbock formula

§4. An elementary formula for the heat kernel on forms.

CHAPTER VI : THE GAUSS-BONNET-CHERN THEOREM

§ 1. Supertraces and the heat flow for forms.

§2. Proof of the Gauss-Bonnet-Chem Theorem.

Bibliography

Notation

Index

360

363

367

371

374

379

391

398

402

407

408

413

422

423

116280

INTRODUCTION

A. There were three main aspects of the theory of diffusions on

manifolds presented in this course: the theory of characteristic

exponents for stochastic flows; the use of the Feynman-Kac formula for

the solution to the heat equation for forms to obtain geometric results,

in particular on the shape of the manifold at infinity; and a technique

for obtaining exact and asymptotic expansions of heat kernels. A proof

of the Gauss-Bonnet-Chern theorem was given as an application of the

third, and also to emphasize the fact that Malliavin calculus is not

needed for probabilistic proofs of the Atiyah-Singer index theorem.

The first two aspects concern long term behaviour and the third

short time behaviour.

Since the participants could not be expected to have a thorough

knowledge of differential geometry, quite a lot of time was devoted to a

quick course in Riemannian geometry via connections on the frame

bundle. Apart from the intuitive understanding this gives to the notion

of parallel translation along the paths of a diffusion, needed for the

Feynman-Kac formula for forms, it allows for a global formalism and is

anyway intrinsically involved in the main example considered in the

discussion of characteristic exponents. Also the fermionic calculus for

differential forms was described, both in order to give a proof of the

Weitzenbock formula: which is a basic result needed to be able to obtain

a Feynman-Kac formula, and in order to give the 'supersymmetric' proof

of the Gauss-Bonnet-Chern theorem. This was taken from [33].

Probabilistically it was assumed that the participants had a

reasonable understanding of stochastic differential equations in !R n ,

driven by Brownian motion, and with smooth coefficients. The existence

theorem for solutions of such equations on manifolds (up to an explosion

time) was proved by embedding the manifold in some !R n and then using

existence results for !R n . See also [1 07].

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B. Not surprisingly, perhaps, it was not possible to cover everything in

these lecture notes during the 15 hours of the course. The main sections

missed out were those relating to the geometry of submanifolds of IR n :

II §7, Ill §3; the section on moment exponents Ill §5; and much of V §2.

C. Acknowledgements. I would like to thank the participants for

their encouragement and helpful suggestions, as well as for their help in

pointing out the numerous errors in the first version of these notes. I

am particularly grateful to Monique Poitier for this. From my point of

view the whole summer school was extremely enjoyable as well as being

very stimulating, and I am very pleased to be able to express my

admiration of Professor Hennequin's efforts which made this possible. I

would also like to thank Steve Rosenberg for permission to use some

unpublished joint work in Chapter IV and to thank David Williams for

pointing out a mistake in my original version of Chapter VI §2C.

This final camera ready version was prepared by Peta McAllister

from the original version typed by Terri Moss: the difficulties they had

from a combination of my writing and changing technology were

considerable, and I owe them a big debt of thanks for their patience,

skill and speed.

Some of the work reported in the course was supported in part by

the U.K. Science and Engineering Research Council.

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CHAPTER 1

STOCHASTIC DIFFERENTIAL EOUA TIONS AND MANIFOLDS

§1. Some notatton, running hypotheses, and basic facts about manifolds

A. We will use invariant notation as much as posssible: if U is open in a

Banach space E and F is some other Banach space then the first derivative of a

function f: U -+ F will be written

Of : U -+ !I...(E,F)

and the second

o2f: u-+ !I...(E,E;F), etc.,

where !I...(E,F) and ll...(E,E;,F) are the spaces of continuous linear and bilinear

maps into F respectively. Sometimes we write

df : u X E-+ F

for

(x,v)-+ Df(x)(v)

with

(df)x for Df(x).

For continuous semi-martingales z1, z2 with values in finite dimensional

normed spaces E1, E2 and for bilinear Bt : E1 x E2 -+ F it will be more

convenient to use

T T

J J ij j j B1(dz1,t, dz2,t) for Li,j Bt d<zl'z2>t

0 0

or for some corresponding expression involving the tensor quadratic variation.

Here the i,j refer to co-ordinates with respect to bases of E1, E2 respectively.

Thus

o2f(x)(dz1,t• dz2,t) = Li,j a2f /oxiaxi d<zi 1• zi2>t

for E = Ei = IRn.

B. Our manifolds M will be coo, connected, and of finite dimension n, unless

clearly otherwise. They will always be metrizable. The manifold structure is

determined by some coo atlas ((Uex, fPex) : ex E A where A is some index set,

(Ucx : ex E A is an open cover, and each fPcx is a homeomorphism of Uex onto

some open subset of IRn, such that on their domain of definition in IRn each

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coordinate change <9o: o <9.13-1 is coo. The pairs (Uo:, <9o:) are C00 charts, as

are any other such which when added to the original atlas still keeps it a coo

atlas. For 0 :$ r :$ oo maps f: M-+ N of manifolds are cr if each ep o f o <9o:-1

is cr on its domain of definition when (Uo:, <9o:) is a chart forM and (Vp, ep)

one for N. The map is a cr diffeomorphism if it is a homeomorphism and both

it and its inverse are cr. The spaces lRn are considered as coo manifolds by

taking ((lRn, identity map)) as atlas, and similarly for open sets U of lRn. A

subset N of [Rn+p is an n-dimensional coo submanifold if there is a family of

coo charts ((UJ3, cp.13): J3 e B) for lRm such that UJ3 covers M and IPJ3-1 (lRn x (O)

= N n U.13 for each .13 e B (writing [Rn+p = lRn x lRP). Then ((U.13 n N, IPJ31UJ3 n N):

J3 e B) forms an atlas for N, making it a coo manifold.

A coo map f : M -+ [Rn+p is an embedding if its image N := f(M) is a coo

submanifold of [Rn+p and f gives a diffeomorphism of M onto N.

C. A tangent vector at x e M can be considered as an equivalence class of

smooth curves

~: (-£,£)-+ M

some £ > 0 with ~(0) = x where ~1- ~2 if d/dt <9o:(~1(t)) and

d/dt ~Po:(~2(t)) agree at t = 0 for some (and hence all) charts (Uo:, ~Po:) with

x e Uo:. The set of all such forms the tangent space T xM to M at x. Any chart

around x gives a bijection

T x~Po: : T xM -+ lRn

which is used to give T xM a vector space structure (independent of the choice

of chart). When M is lRn itself, or an open subset of lRn, the tangent space is

naturally identified with lRn itself. A cr map f : M-+ N of manifolds r ~ 1

determines a linear map

T xf : T xM-+ T f(x)N

(obtained by considering the curves f o ~for example).

The disjoint union TM of all tangent spaces (T xM: x e M) has a projection

"t: TM-+ M defined by "t-1(x) = T xM. This is the tangent bundle: it is a coo 2n­

dimensional manifold with atlas

("t-1(Uo:), T<po:) : o: E A)

where

T<po:(v) = (cpo:(x), T x <9o:(v)) e lRn x lRn

for v e T xM, for ((Uo:, <9o:) : o: e A an atlas of M. Our cr map f : M-+ N

determines a cr-1 map Tf : TM-+ TN where Tf restricts to T xf on T xM. The

assignment of Tf to f is functorial. Note that for U open in lRn we can identify

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TU with U x IRn.

For f: M-+ IR write df: TM-+ IR for v-+ T "t(v)f.

D. When f: M-+ IRn+p is an embedding each T xf: T xM-+ IRn+p is injective, and

Tf: TM-+ IRn+p x IRn+p, namely v-+ (x,T xf(v) for veT xM, is an embedding. This

is used to identify T xM with its image in IRn+p.

Now suppose M is a closed submanifold of IRn+p (i.e. a closed subset,

not necessarily compact): the identity map is an embedding and we make the

above identifications. There is the normal bundle v(M) toM:

v(M) = (x,v) eM x IRn+p: v ~ T xM.

Then v(M) has the obvious projection onto M with fibres vx(M) : x e M linear

spaces of dimension p. By considering the map

P : M x IRn+p -+ T xM

P(x,v) = Px(V) E T xM

where Px is the orthogonal projection of IRn+p onto T xM, use of the implicit

function theorem shows that v(M) is a submanifold of IRn+p x IRn+p.

Next we will construct a tubular neighbourhood of M in IRn+p; this will be

used to save worrying about details in the proof of the existence of solutions

of S.D.E. The exponential map of IRn+p is the map

exp : TIRn+p = IRn+p x IRn+p -+ IRn+p

exp(x,v) = x + v.

Restrict it to v(M) to get lj) : v(M)-+ IRn+p• say. Differentiate at (x,O) to get

T(x,O)IP: T(x,O) v(M)-+ IRn+p_ Now T(x,O) v(M) consists of the direct sum of

T xM (the "horizontal" part) and the tangent space to vx(M) at 0 which can be

identified with Vx(M) itself since it is a linear space. With this identification

T(x,o)IP(v,w) = v + w

so that T(x,O)IP is a linear isomorphism. The inverse function theorem

(applied in charts) implies then that lj) restricts to a coo diffeomorphism of a

neighbourhood of (x,O) in v(M) onto an open neighbourhood of x in IRn+p.

Piecing these together for different x in M we see there is an open

neighbourhood of the zero section (x,O) e v(M) : x e M, which is mapped

diffeomorphically onto an open neighbourhood of M in IRn+P. In particular

(using a coo partition of unity on M) there is a smooth function a : M-+ (0, oo)

such that t.p gives a diffeomorphism lj) :(x,v) e v(M) :I v I< a(x)-+ Na(M) where

Na(M) = UxeM y e IRn+p s.t. I y-x I< a(x). This will be called the tubular

neighbourhood of M radius a. Note that on Na(M) the map giving the distance

squared from M

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y-+ d(y,M)2

is C00 since d(y,M) =I projection on the second factor of ~.p-1(y) 1. Examples

to think of are the spheres sn in 1Rn+1 and the surface of revolution z = (x2 +

y2)-1, for x,y positive, in IR3.

§2. Stochastic differentfal equations on M

A. For a probability space (Q,~,IP) consider a filtration ~t: t ~ 0, assuming

for simplicity that each ~ t contains all sets of measure zero in ~.

Let Y: M x IRm-+ TM be c2 with each Y(x)(-) := Y(x,-) linear from IRm into

T xM· For t 0 ~ 0 let zt, t ~ t 0 be a continuous semi-martingale on IRm and let

u:Q-+M

be ~t -measurable. By a solution to the stochastic differential equation 0

dXt = Y(Xt) o dZt t ~ t 0 (1)

with x1 = u 0

we mean a sample continuous, adapted process,

Xt : Q -+ M t0 s t < s where s : Q-+ [O,oo] is a stopping time such that for any c3 map f : M-+ IR and

stopping timeT with t 0 :S T < s there is the Ito formula

T

f(xT) = f(u) + J T x f(Y(xs) o dzs) \) s

a.s.

where o denotes Stratonovich integral.

(2)

B. The pair (Y,z) will be called a stochastic dynamical system (S.O.S.). Write

(to.s) X Q for (t,w) E [to,oo) X Q: t < s(w). When M = IRn this is equivalent to

the usual definition and there is the basic existence and uniqueness theorem:

Existence and Uniqueness Theorem for IR"

Given z,Y, and u as above forM= IRn there is a unique maximal solution

to (1)

defined up to an explosion time ~ which is almost surely positive. On

(~ < oo, as t t ~(w) so Xt(w)-+ oo in IRn a.s.

Uniqueness holds in the sense that if (Yt: t0 :S t < S is any other solution

starting from u at time t 0 then s s ~ almost surely and x I [t 0 ,s) x Q = y

almost surely. Indeed this is easily deduced from the usual case of an Ito

equation with globally lipschitz coefficients: for R = 1,2, ••• take a C00 map <pR :

122286

IRn-+ IR(> 0) identically one on the ball B(O;R) about 0 radius R and with

compact support. Set YR =<pRY . In its Ito form

dXt = YR(Xt) o dZt

has Lipschitz coefficients and has a solution xR, say, defined for all t ~ t0 ,

starting at u at time t 0 • If SR is its first exit time from B(O;R) then XR' agrees

with xR up to time SR if R' > R , and so a limiting solution x to (1) can be

constructed as required.

C. Existence and uniqueness theorem for M: We will now see why the same

holds for a general manifold M. The straightforward proofs of this tend to be

either tedious or unsatisfying, so we will try to avoid tedium by some

geometrical constructions.

First of all for there to be uniqueness there clearly has to be an ample

supply of c3 functions f. Since we will have to use that fact we may as well

use a strong form of it, namely: Whitney's embedding Theorem. There is a

C00 embedding <p : M-+ IRn+p of M onto a closed submanifold of IRn+p where

p=n+1.

The exact codimension p will not be important, and the proof then, for

some p, is a straighforward argument using partitions of unity and the implicit

function theorem.

Taking such an embedding <p, identify M with its image, so that we can

consider it as a submanifold of IRn+p. It is easy, using C00 partitions of unity

to extend Y to a c2 map Y: IRn+p x IRm-+ IRn+p x IRn+p. However we will do a

more explicit extension below. Since a c2 function on IRn+p restricts to a c2

functlon on M a solution to (1) is a solution to

(I)

in IRn+p (strictly speaking, if Xt satisfies (I) then <p(Xt) satisfies (1) since

f =I o <pis c3). Thus uniqueness holds for (1) since it does for (I). Conversely

any c3 map f : M-+ IR can be extended to a c3 map I: IRn+p-+ IR , so a solution

to (I) which lies on M for all time must solve (1). To prove existence of a

maximal solution it is enough therefore to show that if xt : t 0 :S t < S is the

maximal solution to (1) with Xt = u for u : Q-+ M then Xt lies in M almost 0

surely for all t. In fact it is enough to do that for just one extension Y. In

particular take a tubular neighbourhood of M radius a: M-+ IR(> O) as described

in §1D; chooseR> 0 and let

123287

aR = inf a(x): X eM n B(O;R + 1)) > 0.

Choose smooth)..: IRn+p-+ 1R(2: O) with support in B(O;IR+1) and identically one

in B(O,R), and smooth Jl : [O,oo)-+ 1R(2: 0) with J.l(X) = 1 for I x Is ~ aR2 and,

J.l(X) = 0 for I x I> aR2 •

Let TI : Na(M)-+ M map a point to the nearest point of M to it. Using the

identificatlon tp of Na(M) with part of v(M) this is seen to be C00 • Define

YR: IRn+P x IRm-+ IRn+p by

YR(x)(e) = 0 for x ~ Na(M), e e IRm

YR(X)(e) = J...(X)jl(d(x,M)2) Y(TI(x))(e) for X E Na(M), e E IRm.

Define f: IRn+p-+ IR 2: ·0 by f(x) = J...(X)J.l(d(x,M)2).

Inside B(O;R) the map f is constant on the level sets of d(-,M) while

YR(x)(e) is tangent to these sets. Therefore

Df(x) (YR(x)e) = 0 X E B(O,R), e E IRm

and so any solution Xt: t 0 s t < S) to (r) starting on M satisfies f(Xt) = 0

almost surely until it first exits from B(O;R). It therefore stays on M almost

surely until this time. Since R was arbitrary we are done:

The uniqueness and existence up to an explosion time result of §28 holds

exactly as stated when IRn is replaced by a manifold M.

D. It is now easy to see:

If N is a closed submanifold of M and Y(x)e lies in T xN for all points x

of N then any solution to (1) which starts on N almost surely stays on N

for its lifetime.

Indeed we now know there is a solution to the restriction of (1) toN which

exists until it goes out to infinity on N, or for all time. This solves (1) on M

and is certainly maximal: it must therefore be the unique maximal solution to

1).

E. Usually our equations will be of the form

dXt = X(Xt) o dBt + A(Xt)dt (1)'

where X : M x IRm-+ TM is as Y was before, (Bt: t 2: O) is a Brownian motion on

IRm, and A is a vector field on M, so A(x) e T xM for each x eM. This fits into

the scheme of (1) with Zt = (Bt,t) e IRm x IR = 1Rm+1 and Y(x)(e 1,r) = X(x)(e 1) +

rA(x) when (e 1,r) e IRm x IR. However in this case it is of course only

necessary to assume that X is c2 and A is c1.

m

For an orthonormal base e1, ••. ,em of IRm write Bt as L gr t ero so the r=l

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Brt: t 2: 0, r = 1,2, ... ,m are independent Brownian motions on IR. Let x1, ... ,xm

be the vector fields

XP(x) = X(x)(ep)

Then (2) becomes

§3. An Ito formula

p = 1, ... ,m.

(2)

A. We will need (2) in Ito form. One version using covariant derivatives will

be given below in § 2 of Chapter II. However it will be useful to have a form

which does not depend on a choice of connection for M e.g. when we need to

consider equations on pri nci pal bundles it would be a nuisance to have to

describe some way of covariant differentiation of vector fields on principal

bundles.

For equation (1) and each e e lRm let

( t,x) -+ S( t,x)e

be the flow of the vector field Ye := Y(-)(e) on M, defined on some

neighbourhood of 0 x Min IR x M.

A vector field, e.g. Y e• acts on f : M -+ IR to give

Yef:M-+IR

by Y e f(x) = df(Y e(X) ).

Thus, for our c2 function f

and

d/dt f(S(t,x)e) = df(Ye(S(t,x)e))

= Yef (S(t,x)e)

d2/dt2 f(S(t,x)e) = Ye Yef(S(t,x)e).

At t = 0 this gives linear and bilinear maps which we write

d/dt foS(t,x) I E IL(IRm;IR) t=O

and

d2/dt2 foS(t,x) I E IL(IRm, IRm;IR) t=O

(3)

(4)

Propos1t1on 3A (Global Ito formula) Far a solution xt: 0 :S t < S of (1), if

f: M-+ IR is c2 and Tis a stopping time less than~ then, using Ita integrals,

almost surely:

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T T

f(xr) = f(x 0 ) +J d/dt f•S(t,xr)l (dzr) + ~J d2 /dt2 f•S(t,xr)l (dzr,dzr) (5) 0 t=O 0 t=O

In another form: if e 1, .•• ,em is a base for IRm and

then

T T

f(xr) = f(x 0 ) + J LvP f(xr)dzPr + ~ J L vP Yqf(xr) d(zP,zq>r (6) 0 p 0 p

Proof. If f is c4 the Stratonovich correction term for (2) is

T

~ J L dvPt dzqt where vPt = df(YP(x1)) = YPf(x1), 0 p,q

and (6), and hence (5), follows by calculating dvPt using (2) applied to YP(t).

For f only c2 the easiest way to proceed is to embed M in some IRn+p, extend f

to some c2 function f, extend Y toY as before and write

in Ito form. Then apply the usual Ito formula to f to obtain (6) after

restriction. //

B. For equation (2)'

dXt = X(Xt) o dBt + A(Xt)dt

form (6) becomes

T T

f(XT) = f(x 0 ) + J df(X(Xr)) o dBr + J .A. f(Xr)dr 0 0

where

m

.A.f = ~ L xPxPr + Af. p=l

(6)'

Using the results for IRn+p after embedding M in IRn+p we see the

solutions of (2)' form a Markov process with differential generator .A. •

C. A sample continuous stochastic process Yt : 0 ::s t < S) on M is a semi­

martingale if f(y1) : 0 ::s t < 0 is a semi-martingale in the usual sense

whenever f : M..,. IRis c2, (Schwartz [90]). The above formulae show that our

126290

solutions x1

to (1) are semi-martingales. There is also the converse result,

observed by Schwarz in [90], every continuous semi-martingale y on M is the

solution of some equation like (1): indeed given y take some embedding cp: M-+

IRn+p, some p. Let Zt = cp(yt)· Then z is a semi-martingale, and if P: M x IRn+p

-+ TM is the orthogonal projection map, as in §10, then y is a solution to

dxt = P(xt) o dzt (7)

One easy way to see this is to use the projection TT : Na(M)-+ M of a tubular

neighbourhood as in §1D: then, for Xt = Yt• equation (7) is the differential form

of the equation n(zt) = Zt·

D. For a cl map f: M-+ N of manifolds, stochastic dynamical systems (X,z) on

M and (Y,z) on N are said to be f-related if T xf(X(x)e) = Y(f(x))e for x eM and

e e IRm. The corresponding result for O.D.E. together with equation (5)

immediately shows that it f ts c2 and Cxt : o :::::; t < 0 is a solution to dxt =

X(Xt) o dZt then f(Xt): 0:::::; t < S is a solution to d!:Jt = Y(yt) o dZt on N. For a c3

map f it is immediate from the definition.

§4. Solut1on flows

A. Flows of stochastic dynamical systems were discussed in Kunita's

Stochastic Flow course in 1982 , [67], and I do not want to go into a detailed

discussion of their existence and properties. However I would like to describe

briefly a method which gives the main properties of these· flows rather

quickly, and also mention some rather annoying gaps in our knowledge. The

first problem is to find nice versions of the map

(t,x,w)-+ Ft(x,w) eM we Q

which assigns to x e M the solution to the S.D.E. starting at x at time 0. In fact

throughout this section we assume Zt = (Bt,t) e IRm+l so that we are really

dealing with (1)'.

B. ForM compact, or more generally for Y of compact support, or for

M = IRn with Y having all derivatives bounded it is not difficult to show that

Totoki's extension of Kolmogorov's theorem can be used to obtain a version ofF

such that

(i) For all X EM, Ft(X,-): t ~ 0) solves dXt = Y(Xt) o dZt with F0 (x,-) =X.

(ii) Each map [O,oo) x M-+ M given by

(t,x) -+ F t(x,w) is continuous.

This was the method used by Blagovescenskii and Freidlin, for example see

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[43], [59], [9], [67], [78]. The extension of Kolmogorov's theorem used is:

Let (M,d) be a complete metric space. Suppose for x : [0,1] x ••• x [0,1]

(p-times) -+ t 0 (Q, ~;M) there exists ex, j3, 6 > 0 such that for all o > 0 and

s,t in [0,1]P

IP d(xs,Xt) > & s j3o -ex Is-tiP+'¥

then x has a sample continuous version.

The necessary estimates are most easily obtained by embedding Min some

IRn+p and extending Y as before.

C. For M compact it is possible to obtain differentiability, diffeomorphism,

and composition results by considering an induced stochastic differential

equation on the Hilbert manifold of Hs diffeomorphisms of M for s > ~n + 3: the

solution of this equation starting at the identity map being a version of Ft, see

[43]. Rather than discuss the Hilbert manifold structure of these groups of

diffeomorphisms it is possible to embed M in some IRn+p and extend the S.D.S.

over IRn+p as before, to have compact support. A now for the extended system

will restrict to one for the system of M. For the extended system we consider

the space of diffeomorphisms of class Hs of IRn+p which are the identity

outside of a fixed bounded domain U, containing the support of the extended

system. We will describe this rather briefly, see [240] for details.

Suppose therefore there is the system on IRn

dXt = Y(Xt) o dZt

where Y has compact support in U, for U open, bounded, and with smooth

boundary.

For s > n/2 set

Hsu(IR 0 ; IRP) = f e Hs(IRn;IRP) with suppf c U ) where Hs(IRn;IRP) is the

completion of the space C000 (1Rn; IRP) of C00 functions with compact support

under < >s where

(f,g)s = ~ JIRn (Dexf(x), oexg(x))dx

where the sum is over multi-indices ex = (ex 1 , ... ,exn) with oex =

I I ex1 exn a ex /(ox 1 .•• oxn ). Because s 2: n/2 the evaluation map x -+ f(x) is

continuous on Hs and so Hsu is a well defined closed subspace of Hs.

For s > n/2 + 1 set

.t'lsu = f : IRn-+ IRn s.t. f is a c1 diffeomorphism and 1d-f e Hsu(IRn;IRn) where

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1d refers to the identity map. Since diffeomorphisms are open in the c1

topology llsu is an open subset of the affine subspace 1d + Hsu(IRn;IRn). It is

therefore a Hilbert manifold with chart f-+ f - 1d. Moreover

1. llsu is a topological group under composition

2. For hE llsu right multiplication

Rh : llsu-+ llsu

f-+foh

is C00 •

3. Fork= 0,1,2, ••• composition

<pk : lls+ku x llsu -+ llsu

(f ,h) -+ f 0 h

is ck.

4. For k = 1,2, .•. inversion h-+ h -1 considered as a map

:Jk :lls+ku-+ llsu

is ck.

Now define the right invariant stochastic dynamical system

dht = Y(ht) o dzt

on llsu, always assuming s > n/2 + 1, by taking

Y(h) : IRm -+ T h llsu ==< Hsu h e llsu

to be

Y(h)(e) = <p2(Y(-)(e),h)

treating x-+ Y(x)(e) as in Hus+2. Thus Y is c2 by (3) above. We see

Y(1d)(e) = Y(-)(e)

and

Y(h)(e) = DRh(ld)(Y(Id)e)

so that Y is right invariant.

Proposltjon 4C. The solution ht to dht = Y(ht) o dzt starting from 1d

exists for all time, and Ft(-,w) = ht(w)(-) gives a flow for our equation on IRn

lying in J:lSu· In particular there is a C00 version of this flow, such

that t-+ F1(-,w) is almost surely continuous into the C00 topology.

Proof: A maximal solution certainly exists up to some predictable stopping

tlme s say, with s > 0 almost surely: the theory for equations of this type on

open subsets of a HHbert space goes through just as in finite dimensions,

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(provided one always uses uniform estimates, i.e. uses basis free notation).

Choose a predictable stopping t1 me T with 0 :S T :S s. By the right 1 nvari a nee of

the system, for h' e llsu

(s,w)-+ hs(w).h'

is a solution starting at h', so

(s,w)-+ hs-T(w)<eT(w))hT(w)(w)

where eT is the shift, is equivalent to hs(w) for s > T(w). Thus

s(w) 2::: T(w) + T(eT(w)).

Iterating this

.. s(W) 2::: k T (ekT(w)).

co Since Toek-rlk;O is an i.i.d. sequence we see s = oo almost surely. To see that

ht(w)(x0 ) is a solution to the equation on IRm starting at x0 observe that the two

equations are f-related for f the evaluation map f(h) = h(x0 ), and use the result

of §30. Finally since s > n/2 + 1 was otherwise arbitrary we see that almost

surely s-+ hs(w) lies in the space of continuous maps into

a J)SU With 1ts induced topology: but this is just a subset Of the space Of

C00 diffeomorphisms, with its relative topology. II

By embedding we deduce the existence of a C00 flow of diffeomorphisms

F1(-,w): M-+ M continuous in t into the C00 topology when M is compact.

Results like those of Bismut and of Kunita on the stochastic differential

equations for the inverses of the flows and compositions are easily obtained

via the inflnite dimensional Ito formula, see [24].

D. For non-compact manifolds M we can embed M and approximate the extended

S.O.S. by equations with compactly supported coefficients as in §1F. This

gives [66], [24a] Kunita's results on the existence of partial flows:

There is an explosion time map~ : M x Q-+ (O,oo] and a partially defined

flow F for dx1 = Y(xt) • dzt such that if

M(t)(w) = x EM: t < ~ (x,w) we Q

then for all w e Q

(i) M(t)(w) is open in M i.e. S(-,w) is l.s.c.

(i1) Ft(w) : M(t)(w)-+ M is defined and is a C00 diffeomorphism onto an open

subset of IRn

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(iii) For each x in M, )(x) is a stopping time and (Ft(X,-) : o :s; t < )(x)) is

a maximal solution. Moreover if K is compact in M, if S(K)(w) = inf

()(x)(w):x e K) then on )(K)(w) < oo, for any x0 eM, sup d(x0,Ft(x,w))-+ xEK

oo almost surely as t t )(K)(w).

(iv) the maps-+ Fs(-)(w) is continuous from [O,t] into the C00 topology of

functions on M(t)(w).

When we can choose s = oo the system is said to be strongly complete, or

strictly conservative. The standard example of a system which is complete

but not strongly complete is

dXt = dBt

on M = !R2 - (0). It is complete since Brownian motions starting outside of 0

in IR2 almost surely never hit 0. However the flow would have to be

Ft(X,W) =X+ Bt(W)

so that

s(x,w) = inf (t : Bt(W) = -X).

Using the fact that the solution to a complete system exists for all time

even when starting with a given random variable x0 : Q-+ M (independent of the

driving motion of course) it follows [27] that given completeness

J1 ® IP((x,w) : s(x,w) < oo) = 0

for all Borel measures J1 on M. In particular for almost all w £ Q each open set

M(t)(w) has full measure in M.

Even when the system is strongly complete the maps Ft(-,w) may not be

surjective: the standard example is the 2-dimensional Bessel process on M = (O,oo), see [43], [44], [59]. It seems difficult to decide when strong

completeness holds. It depends on Y, not just on the generator A: as observed

by Carverhill the Ito equation

dZt = (Zt/1 Zt I) dBt

on a::: - (O) is strongly complete (using complex multiplication for 8 a

Brownian motion on a:::::::: IR2), yet its solutions are just Brownian motions on

IR2 - (0). A general problem is therefore to find conditions on A and M

which ensure that there exists a choice of Y , giving a diffusion with

generator A , which is strongly complete. Anticipating some concepts from

the next chapter: for a complete Riemannian manifold M does a lower bound

on the Ricci curvature ensure strong completeness of the canonical SDS on

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OM? What conditions on M ensure that it admits a strongly complete SDS with

A = ~~ ? More generally is strong completeness of Y implied by the existence

of a uniform cover for Y in the sense of [43], [24a]?

E. Assume now that the system is complete. Even if it is not strongly

complete we can formally differentiate solutions in the space directions, to

get "derivatives in probability" rather than almost sure derivatives, [43]. This

was done forM= IRn in [57] to get L2-derivatives given strong conditions on Y.

These agree with the almost sure derivatives in the derivatives of the flow

F1(-,w) where the latter exists, and we will write them here as T x F1(-,w): 0

T x0

M--. T Xt(w)M, as if the flow did exist.

For veT x M set Vt = T x Ft(v,-). Then, either by embedding in some !Rn+p 0 0

or by arguing directly, (vt: t 2: 0) satisfies an equation

dv1 = &Y(v1) o dzt

on TM, where now &Y: TM x IRm--. T(TM) and is given by

&Y(v)e = ex TYe(v)

for Ye = Y(-)e : M -+ TM and ex : TTM -+ TTM the involution which over a chart

which represents TU as U x IRn and so TTU as (U x IRn) x (IRn x IRnl is given by

(x,u,v,w)-+ (x,v,u,w).

Thus in these co-ordinates

o Y( (x,u))e = (x,u, Y(x)e,DY e(x)( u)).

132296

CHAPTER II

SOME DIFFERENTIAL GEOMETRY FOR PRINCIPAL BUNDLES AND CONSTRUCTIONS

OF BROWNIAN MOTION

§1. Connections on principal bundles and covariant differentiation

A. A Lie group G is a C00 manifold with a group structure such that the maps

GxG-+G and

are C00 • Standard examples include the circle s1, the three sphere s3 (the

multiplicative group of quaternions with unit norm), orthogonal groups O(n)

which have SO(n) as connected component of the identity, and non-compact

groups (IRn, +)and GL(n).

A right action of G on a manifold M is a C00 map M x G-+ M

usually written (x,g)-+ x.g such that x.1 = x (for 1 the identity element), and

(x.g1)·92 = x.(g1.g 2 ). Examples are the action of G on itself by right

multiplication. The natural action of GL(n) on IRn is a left action, defined

similarly. Note that x-+ x.g is a diffeomorphism of M, which we will write as

Rg: M-+ M, with Lg : M-+ M for a left action.

Let g be the tangent space to G at 1. Then the left action gives a

diffeomorphism (trivialization of TG)

y: G X g-+ TG

Y(g)(v) = T 1 Lg(v).

Thus a semi-martingale z on g gives an S.D.E. on G.

The map v-+ Yv := Y(-)(v) gives a bijection between g and the space of

left-invariant vector fields on G. For any two vector fields x1, x2 on a

manifold M there is another vector field [x1, x2], the Lie bracket, determined

by [xl, x2]f = x1x2r- x2x1r for f: M-+ IR a c2 function. This gives a Lie

algebra structure. For a Lie group the bracket of two left invariant vectors

remains left invariant, so there is an induced Lie bracket on g s.t.

Y[v1,v2] = [Yv1,Yv2].

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Let <5(t): t e IR) be the curve in G with <5(0) = 1 and d<5/dt = Yv(<'(t)).

It is a 1-parameter subgroup and usually written o(t) = Exp(tv). This

determines

Exp: g~ G

b!J v ~ Exp v, not to be confused with other exponential maps (e.g. in § 1D and

§28 below).

B. A principal G-bundle over M is a map TT: 8 ~ M of differentiable manifolds

which is surjective, where B has a right G-action s.t. n(b.g) = n(b) all be B,

g e G, and such that there is an open cover (Ucx: : ex: e A of M for which there

exist C00 diffeomorphisms (local trivializations)

ecx:: n-1cucx:) ~ Ucx: X G

of the form

with

ecx:,x(b.g) = ecx:,xCb ).g.

X = TT(b)

The simplest example is the product bundle M x G with obvious right

action. An important example is the full linear frame bundle of M , TT : GLM

~ M, where GLM consists of all linear isomorphisms u : IRn ~ T xM for some x e

M, with TT mapping u to the relevant x. The right action is just composition

u.g(e) = u(ge) for e e IRn. Each element u is called a frame since it can be

identified with the base (u1•···•ur1) of T xM where Up = u(ep) for e1, ... ,en the

standard base for IRn.

Given principal bundles n i : Bi ~ M with groups Gi for i = 1,2 a C00

homomorphism is a C00 map h: s 1 ~ s2 such that h(b 1.g 1) = h(b 1).h 0 (g 1) for

some smooth group homomorphism h0 : G1 ~ G2. Thus every principal bundle

is by definition locally isomorphic to the trivial(:= product) bundle, (with h0 =

identity map).

C. For a principal G-bundle n : 8 ~ M the tangent space TB has a naturally

defined subset: the vertical tangent bundle, or bundle along the fibres,

VTB = v e TB: Tn(v) = 0.

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A connection on 8 is an assignment of a complementary "horizontal" tangent

bundle, HT8, invariant under the action of G. One way to do this is to take a g­

valued 1-form ro i.e.

ro : T8-+ g

is smooth, and each restriction rob: Tb8-+ g, bE 8 is linear, with

(i) roo TRg = ad(g-1) oro g E G (8)

where ad(g-1): g-+ g is the adjoint action, namely the derivative at 1 of the

map G-+ G, a-+ g-1ag, and

(ii) ro(A*(b)) =A bE 8, A E g (9)

where A* is the (vertical) vector field on 8 defined by

A *(b) = d/dt (b. exp tA) lt=o·

Such an ro is called a connection form. Given such,one can define a

horizontal tangent bundle by

HT8 = v E T8: ro(v) = 0).

Then

(a) Tb8 = HTb8 EB VTb8

and

(b) TRg(HTb8) = HTb.g8

each bE 8

bE 8, g E G.

Conversely given HT8 satisfying (a) and (b) and smooth there exists a

connection form inducing it. It is easy to construct connections by partitions

of unity: but without additional structure there is no canonical choice.

For each trivialization (Uo:, eo:) there is a local section

so:: uo:-+ n-1cuo:) c 8

so:(x) = e-1o:,xC1)

This can be used to pull back a connection form ro to a g-valued 1-form roo: =

so:*(ro) given by the composition roo:= rooTso:: TUo:-+ g. For a connection on

GL(M) the components of this will give the Christoffel symbols: indeed a chart

(Uo:,fPo:) for M determines a trivialization which maps u to (T x fPo:) o u: !Rn-+

!Rn for a frame u at x. Then so:(x) = (T xfPo:)-1: !Rn-+ T xM. Define

f' : fPo:(Uo:)-+ IL(IRn;g) = IL(IRn;L(IRn;IRn))

by

135299

f'C<J'cx(x))v = scx*(ro)(T x\Pcx - 1(v)) v E IHn

giving the classical Christoffel symbols rijk

ri jk(Y) = (r(y)( ej)(ek),ei >

(10)

( 11)

where e1, ... ,en is the standard base for IRn. The connection is torsion free if

r(y)(v1)(v2) = r(y) (v2)(v1) or equivalently if rijk = r\j·

§2. Horizontal lifts, Covariant derfvatives, geodesics and a second form of the

Ito formula.

A. A connection for n: 8-+ M determines a horizontal lifting map

Hb : T n(b)M-+ T b8

which is the inverse of the restriction of Tbn to HTbB, a linear isomorphism.

It also gives a way of horizontally lifting smooth curves in M to curves in 8; a

smooth· curve <5 in 8 is horizontal if cr(t) E HT8 for all t or equivalently if

w(cr(t)) = o all t:

For a piecewise c1 curve <5: [O,T)-+ M and b0 E n-1(<5(0)) there exists a

unique horizontal curve <5- in 8 with <5-(o) = b0 and n(<5-(t)) = <5(t) for all

t (i.e. <5- is a lift of <5).

In fact given a trivialization (Ucx,ecx), while <5(t) is in Ucx, for <5- to be a

lift ecx(<5-(t)) has to have the form

ecx(<5-(t)) = (<5(t),g(t)) e Ucx x G

and then it will be horizontal if and only if

d/ dt g( t) = - TRg(t) (fi\x< <5( t))(cr( t))) ( 12)

so that local existence and uniqueness follow immediately from standard O.D.E.

theory on G. To see how (12) arises note that the axioms for 6i imply that for

(v,A) in T(x,a) (Ucx x G)

ffio(T9cx)-1(v,A) = ad(a-1)fficx(x)v + TL -1A (13) a

Note that by uniqueness and invariance, if g E G the horizontal lift of <5

starting at b0 .g is just t-+ <5-(t)g.

The only real difficulty in extending this construction to lifts of semi­

martingales Yt (e.g. by treating (12) as a Stratonovich S.D.E. when <5(t) is

replaced by y1) is to make sure the lifted process does not explode before the

136300

original one does see (77], [34], and [43] p. 175. However we will not need

this.

B. For simplicity restrict attention now to an affine connection for M i.e.

a connection on GLM. Given a piecewise c1 curve

cs: [O,T)--. Manda vector v0 E T cs(o)M define the parallel translate

llt(v0 ) E Tcs(t)M 0 :s t :s T

of v0 along cs by

II 1(v 0 ) = cs-(t)b 0 -1(v 0 ) (14)

where cs-(t) is the horizontal lift of cs through b0 E n-1(cs(O)), the result being

independent of the choice of b0 •

Then lit: T cs(o)M--. T cs(t)M is a linear isomorphism.

If we now have vector field W along cs i.e. W : [O,T] --. TM with W(t) E

T cs(t)M for each t, define its covariant derivative along cs by

DW lot= lit dldt (//t-1 W(t)) (15)

Thus W is parallel alongcs i.e. W(t) =lit W(O) iff DWiot = o.

Over a chart (Uw~Pcx) forM, with induced trivialization of

n-1cucx), using the same notation as for (12)

T cs(t) ~Pcx (DWiot) = g(t) dldt (g(t)-1 v(t))

where v(t) = T cs(t) ~Pcx (W(t)), and so by (12) the local representative

T cs(t) ~Pcx (DWiot) is given by

dvldt + r(csa(t))(cra(t))(v(t)) (16)

for cra(t) = G>a(cr(t)) or, if olox1, ... ,oloxn denote the vector fields over Ucx

given by T x~Pcx (oloxi) = ei, the i-th element of the standard base of !Rn and if

W(x) = L.: wi(x) oloxi, and

DWiot = L.: (DWiot)i oloxi etc.,

then

(DWiot)i = dWildt + rijkCcscxCt))(cr(t)i)wk(t)

summing repeated indices.

(17)

By definition a curve cs in M is a geodesic if its velocity field cs is parallel

along cs i.e.

Dlot cr = o. Substitution of this into (17) gives the classical local equations. The existence

137301

theory for such equations shows that for each v0 e T x M there exists a unique 0

geodesic '¥0): 0 :5 t < t 0 for some t 0 > 0 with '¥(0) = x0 and y(O) = v0 • If we

can take t 0 = oo for all choices of v0 so geodesics can be extended for all time

the connection is said to be (geodesically) complete: (note that no metric is

involved so far). The geodesic y above is often written y(t) = expx tv 0 and 0

there is the exponential map defined on some domain .D of TM

exp : .D -+ M x M

exp v = (x, expxv)

when v e T xM· A use of the inverse function theorem shows that there is an

open neighbourhood .D 0 of the zero section Z[M] = image of Z : M-+ TM given

by Z(x) = 0 e T xM, such that exp maps .D 0 diffeomorphically onto an open

neighbourhood of the diagonal in M x M. In particular each expx : T xM-+ M is a

local diffeomorphism near the origin. The inverse determines a chart (U, cp)

around x by q> = expx-1: U-+ TxM ~ IRn. These are normal (or geodesic, or

exponential) coordinates about x. If '¥ is a geodesic in M from x then its local

representative in this chart, cp(y(t)) : 0 :5 t < t 0 , say, is just the 1/2 ray

segment (tv : 0 :5 t < t 0 , where v = 6(0). In particular we see from (15) and

(16) that for a torsion free connection, at the centre of normal coordinates

the Christoffel symbols (for that coordinate system) vanish.

C. Let IL(TM;TM) = U IL(T xM;T xM). It has a natural C00 manifold xeM

structure with charts induced by the charts of M , and a smooth projection

onto M , as do the other tensor bundles e.g. the cotangent bundle T*M = U IL(T xM;IR), the exterior bundles AP TM, and the bundles of p-linear maps

IL(TM, ••• ,TM;IR) ~ ®P T*M.

Note that a frame u at x determines an isomorphism

PI!... (u) ; IL(IRn;IRn)-+ IL(T xM; T xM)

PI!... (u)(T) = uTu-1

and similarly for the other buntjles mentioned:

p*(u): IRn*-+ T*xM given by p*(u)(.t) = ,tou-1,

and also

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and

P®(u)T = T(u-1(-), ... ,u-1(-)).

These equations also determine representations of GL(n) on O...(IRn,IRn),

A TM, etc. which will also be denoted by pn._, PI\• etc.

D. A vector field A on M determines a map

A~: GLM ~ !Rn

by

A-(u) = u-1 A(n(u))

Similarly a section B of O...(TM;TM), i.e. a map B:M ~ O...(TM;TM) such that B(x) E

O...(T xM;T xM) each x, gives

8~ : GLM ~ O...(!Rn; !Rn)

by

(18)

etc.

The covariant derivative VA of A is the section of O...(TM;TM) defined by

V A(x)(v) = u dA ~(v~) (19)

where v- is the horizontal lift Huv of v to HT uGLM, for v E T xM· This is often

written VA(v) or V vA.

Covariant derivatives of other tensor fields e.g. Sections B of O...(TM;TM)

are defined similarly: VB is the section of O...(TM;O...(TM;TM)) given by

VB(x)(v) = pn._(u) dB"'(v .... ) E O...(T xM;T xM) (20)

for v"' as before. In particular the higher order covariant derivatives are

defined this way, e.g.

v2A = V(VA)

is a section of

O...(TM;O...(TM;TM)) ~ O...(TM,TM;TM).

For a chart (Ucx, ~Pcx) forM around a point x, using the induced

trivialization of n-1cucx) our tensor field C, say, when lifted looks like a map

C' on Ucx X GL(n) given by c'(x,g) = p(g)-1 C'(x) where C' is c in our coordinate

system, and p is the relevant representation, e.g. p(g) = g for vector fields, p

= Pn... etc. In these coordinates v- = (v,- r(x)(v)) so VC(x)(v) is given by

dC'(v) + drp(f'(x)(v))C'(x) (21)

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where diP means the differential of p at the identity

-(r(x)v)*.

In particular if our vector field A is given over Uo: by

A= I Ai a;axi, etc.

then, summing repeated suffices, if Xo: = <po:(x)

[VA(v)]i = dAi(v) + rijk(Xo:)(vi)(Ak(x))

= (oAi;axD vi+ rijkCxo:)(vi)(Ak(x))

e.g. d1p*(r(x)(v) =

(22)

(where formally aAi;axi: Uo: -+ IR means the result of acting on Ai by the

vector field o/oxi : in practice everything is transported to the open set

<po:(Uo:) of IRn in order to do the computations so that oAi;axi is computed as

"o/oxj A i ( <po:(x 1, ••• ,xn))" in the sense of elementary calculus).

Comparing (17) and (22) one sees that if V is a vector field taking value v

at the point x, and if o is an integral curve of V, so cr(t) = V(O(t)), with o(O) = x then

DA/otlt=O = V A(v) = V vA (23)

Note that if V is a vector fleld we can form a new vector field VyA or

VA(V) by

V yA(x) := VA(V(x))

we see from (22), or by working at the centre of normal coordinates that for

a torsion free connection

(24)

D. Covariant differentiation behaves similarly to ordinary differentiation.

For example if o: is a 1-form (i.e. a section of T*M) and A is a vector field

ihen for v e T xM

d(o:(A(·)))(v)"' V vo:(A(x)) + o:(V vA(x)) (25)

One way to see this is to write o:(A(n(u))) = (o:x o u) o u-1A(n(u)) for

u e GLM, x = n(u). Then differentiate both sides in the direction Hu(v).

E. Using the notation of §3A of Chapter I

ct2/dt2 f(S(t,x)e) = d/dt df(Ye(S(t,x)e)) = d(df(Ye(-))(d/dt S(t,x)e)

which at t = 0

= V(df)(Ye(X),(Y9 (x)) + df(VYe(Ye(x)))

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by (25). Thus for any affine connection, (5) can be written

T

f(xT) = f(xs) + J df(X(xr)dzr) s

T

+ ~ J V(df)(Y(xr)dzr)(Y(xr)dzr) + '~f(VY(Y(x)dzr)dzr) (26)

s

which for equation (2)', gives the generator A in the form

Af = ~ ~ V(df)(XP(x))(XP(x)) + df(VXP(XP(x)))) + df(A(x)). p;l

(27)

Non-degeneracy of the S.D.S. (2') i.e. surjecti vi ty of each

X(x): IRm-+ T xM is equivalent to ellipticity of Af: the symbol of A is just

E XP ® xP as a section of TM ® TM, or X(·) oX(-)* as a section of

ll...(T*M;TM)).

§3. Riemannian metrics and the Laplace-Beltram1 operator

A. A Riemannian metric on M assigns an inner product < , >x to each tangent

space T xM of M, depending smoothly on x. Over a chart (Ucx:,<pcx) if

u = ui a;axi, v =vi a;axi are tangent vectors then define the n x n-matrix G(x)

= (gij(X)]i,j by

(28)

The inner product determines a metric d in the usual sense on M,

compatible with its topology, by letting d(x,y) be the infimum of the lengths of

all piecewise c1 curves from x toy, where the length .t(<5) is

b

.8(<5) = J lcr(t)l<5(t) dt a

1

for Julx = <u,u>2x as usual, and <5 is defined on [a,b]. The Riemannian manifold

(i.e. M together with<, >x: x eM) is (metrically) complete if it is complete

in this metric.

For a submanifold M of IRm the standard inner product of IRm restricts to

141305

an inner product < , >x on each T xM considered as a subset of IRm, thereby

determining a Riemannian structure on M. It is a highly non-trivial result, the

Nash embedding theorem, that for every Riemannian metric on a manifold M

there is an embedding into some IRm such that the induced metric agrees with

the given one i.e. an isometric embedding. (In general a smooth map f : M-+ N

of Riemannian manifolds is isometric if (T xf(u), T xf(v))f(x) = (u,v>x for all

x eM and u, veT xM; it is an isometry if it is also a diffeomorphism of M onto

N. Thus an isometric map need not preserve distance.)

B. Given such a metric one can consider orthonormal frames: these are

isomorphisms

u : IRn-+ T xM

preserving the inner products, (u(e),u(e')>x = (e,e'>IRn· The space OM of such

frames is a subset of GLM, and keeping n : OM-+ M to denote the projection it

forms a principal bundle with group O(n): it is a subbundle of GLM in the

obvious sense.

A connection on OM is called a Riemannian connection. w will take values

in the Lie algebra o(n) of O(n) which can be identified with the space of skew­

symmetric n x n-matrices. It can be extended over all of GLM by the action of

GL(n) on GLM, insisting on condition (i) for a connection form (or (b) for the

corresponding horizontal subspaces). Thus it determines a connection on GLM

and so local coordinates have associated Christoffel symbols, which can be

used to compute covariant derivatives.

An important point is that for this induced connection on GLM, given a

curve <5 in M, the horizontal lift <5"" of <5 to GLM starting from an orthonormal

frame stays in OM and is the same as the horizontal lift for the original

connection on OM. An immediate consequence (from the definitions, equations

(14) and (15)) is that parallel translation preserves inner products:

(//tV• 1/tv')d(t) = (v,v')d(o) (29)

for v, v' e T <5(o)M, and for vector fields W, W' along <5

d/dt (W(t),W'(t))o(t) = (DW /at, W'(t))d(t) + (W(t), DW'/ot)o(t). (30)

Consequently, by (23), if W 1> w2 are vector fields and v e T xM then

d(W 1(-),W2(-))(-) (v) = (V W1(v), W2(x))x + (W 1(x), VW2(v))x (31)

C. The metric gives an 1 denti fi cation of T xM with its dual T x *M by

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v -+ v • = <v ,->x- In local coordinates (Uo:•<t'cx) let ~Po:(Y) = (x 1(y), ••. ,xn(y)) for y

E Uo: then dyx1, ... ,dyxn) form the dual basis to o/ox1, .•• ,o/oxn, (strictly

speaking eva 1 u ate d at y). If v = vi a 1 ax i at y then v• = vi d y xi where

vi = gij(y)vi (32) #

Write .2-+ 2 for the inverse of this isomorphism also.

By choosing the vector field A such that for given x E M and v e T xM, A(x) = v and \1 A(x) = 0, equation (25) shows that for a 1-form ex

(33)

Similarly \1 v commutes with the 'raising and lowering of indices' on other

tensor fields.

The gradient, grad f, or \lf, of a c1 function f: M-+ IR is the vector field #

(df) so

<Vf(x),v>x = df(v) (34)

all veT xM· In local coordinates \lf(x) = \lf(x)i a;axi where

\lf(x)i = gii(x) of ;axi (35)

where [gii(x)]i,j is the inverse matrix G(x)-1 to [gij(x)].

D. There will be many Riemannian connections for a given metric. However

it turns out that there is a unique one which is also torsion free. This is

called the Levi-Civita connection. It can be defined in terms of the

Christoffel symbols by

(36)

It is this connection which is usually refered to when considering

covariant derivatives etc. for Riemannian manifolds.

E. A Riemannian metric determines a measure on M, temporarily to be

denoted by Jl, such that if (Uo:, ~Po:) is a chart then the push forward Jlo: of JliUo:

by ~Po: is equivalent to Lebesgue measure on the open set <t'o:(Uo:) of IRn with

Jlo:(dx) = .[det G(<po: -1(x)) A(dx) where A is Lebesgue measure and G is the

local representative of the metric. We shall usually just write dx for Jl(dx) or

A ( dx) and write go:(x) or g(x) for det G(x) . Note -./g(x) = ldet T yiP ex - 11 for

y = ~Po:(x), where 'det' refers to the determinant obtained by using < , >x and

143307

< ' >IRn·

For a cl vector field A on M, the divergence, divA: M-+ IR, is given by

div A(x) = d/dt det T x Ft I (37) t=O

where

(t,x)-+ Ft(X) EM

is the solution flow of A, on its domain of definition in IR x M. It represents

the rate of change of volume by the flow. It is given by

div A(x) =trace VA(x). (38)

From (37), using the change of variable formula for Lebesgue measure one gets

the divergence theorem

J div A(x) dx = 0 (39) M

forM compact, and more generally. Since, by (22), iff: M-+ IR,

div fA(x) = (Vf(x),A(x)>x + f(x) div A(x) (40)

we see from this that div and -V are formal adjoints.

The Laplace-Beltrami operator fl. on c2 functions f: M-+ IRis defined by

M = div Vf

or equivalently

M =trace Vdf = l: V(df)(e;)(ei)

where e1, ... ,en are orthonormal. It determines a self-adjoint operator 6. on

L2(M;IR), [52], [91]. In local coordinates it has the formula

M(x) = gii(x) o2f;axioxi- gii(x) rkij(x) af;axk (41a)

and 1 . 1 . . .

M(x) = g(x)-2 o/oxl (g(x)2 gll(x) af ;axl (41b)

which are easily seen using (21) and (38) for (41a), and (35) and (37) for

(41b).

§4. Brownian motion on M and the stochastic development

A Let M be a Riemannian manifold with its Levi-Civita connection. By a

Brownian motion on M we mean a sample continuous process (Xt : 0 :s t < n, defined up to a stopping time, which is Markov with infinitesimal generator ~6..

From the Ito formula (27) a solution of

144308

dXt = X(Xt o dBt + A(Xt)dt

is a Brownian motion 1f and only if

(i) X(x) : IRm-+ T xM is a projection onto T xM for each x in M i.e. X(x) o X(x)* = identity; and

(ii) A(x) =- ~ L VXP(XP(x)) p

When (i) but not (ii) holds we say that (Xt : 0 :S t < s is a Brownian motion

with drift. The drift is the vector field x-+ A(x) + ~ L VXP(XP(x)). p

B. Note, again from (27) for an arbitrary affine connection, that in general

the generator A for our solution is elliptic if and only if each X(x) is

surjective (in which case the S.D.E. is said to be non-degenerate). In this case

each X(x) induces an inner product on T xM , the quotient inner product, and so

determines a Riemannian metric on M. Thus the solutions to a non-degenerate

S.D.E. are Brownian motions with drift for some (uniquely defined) metric on

M, and equivalently any elliptic A can be written as ~.1. + B for some

first order operator (i.e. vector field) B. Even working on IRn, if one wishes

to deal with elliptic generators A, the differential geometry of the associated

metric will not in general be trivial and can play an important role.

C. Although there always exist coefficients X and A satisfying (i) and (ii)

there is no natural choice which can be applied to general Riemannian

manifolds. However there is a canonical S.D.E. on the orthonormal frame

bundle OM toM, and it turns out that the solutions to this project down to give

Brownian motions on M. The construction, due to Eells and Elworthy, is as

follows:

Define X: OM x IRn-+ TOM by

X(u)e = Hu(u(e)). (42)

For given u0 in OM let (ut: 0 :S t < S be a maximal solution to

dut = X(ut) o dBt (43)

where (Bt: 0 :S t < oo is Brownian motion on IRn, so now m = n. Set x1 = TT(Ut):

Theorem 4C (xt : 0 :S t < S is a Brownian motion on M, defined up to its

ex~losion time.

145309

Proof Suppose g : M ~ IR is c2. Set f = g o TT : OM~ IR. With the notation of the

Ito formula (5), §3A, for e e IRn and u e OM set

'¥t(u,e) = n(S(t,u)e).

Then S(-,u)e is the horizontal lift of l_(u,e) through u. Since

'Yt(u,e) = Tn(X(S(t,u)e)(e))

= (S(t,u)e)(e) E T '¥t(U,e)M

(44)

'Yt is parallel along '¥t and sot -t 6t(u,e) is a geodesic in M, see §2B; also to=

u(e). Thus

and

d/dt f(S(t,u)e)lt=O = d/dt g(lt(u,e))lt=O

= dg(u(e))

ct2/dt2 f(S(t,u)e)lt=O = d/dt (dg(tt(u,e)))lt=O

= V dg(tt(U,e))lt=O + dg(D/ot 'Yt(U,e)lt=O)

= V dg(u(e))

since D/ot 'Yt = 0.

Thus by (5)

T T

g(xT) = g(x0 ) + J dg(ut o dBt) + ~ J 6g(xt)dt ( 45).

0 0

That Xt is a Brownian motion can now be deduced from the martingale

problem method, [92]. Alternatively, to show that Xt has the Markov property

first prove that the di stri buti ons Of Xt : 0 :$ t < s dO not depend on the point u0

e n-1(x0 ) in OM: having done this the Markov property is easily deduced from

that of the solutions of our S.D.E. on OM, (e.g. see [43], §5C of Chapter IX). To

see the lack of dependence up to distribution on u0 observe that if u'0 e n-1(x0 )

then there exists g e O(n) with u'0 = u0 .g. By the equivariance under O(n) of the

horizontal tangent spaces, condition (b) of §1C, ut·9: 0 :$ t <~)satisfies

du't = X(u'1) o dB't

where B't = g-1(Bt)· By the orthogonal invariance of the distributions of

Brownian motion, B't is again a Brownian motion and Ut·9 has the same

distributions as the solution u't of du't = X(u't) o dBt from u' 0 • Since TT(Ut.g) =

146310

n(ut) the required invariance for Xt follows.

The maximality of Cxt: 0 s t < ~) follows from that of Cut: 0 s t <~):if

Ut-+ oo in OM as t-+ ~(w) so does n(ut) since each n-1(x) is compact. II

D. Since each X(u)e E T uOM , the considerations of §2A suggest calling the

solution Cut: 0 s t < ~ the horizontal lift of the Brownian motion

Cxt:O s t <~from u0 : in fact it is easy to see that it locally satisfies the local

equations (12) considered as a Stratonovich equation with x1 replacing 6 1•

We can then define parallel translation along the sample paths of our

Brownian motion

llt(W): T x0

M-+ T Xt(w)M

by

WEQ (46).

Also if CWt: 0 s t <~)is a vector field along Cxt: 0 s t <~)i.e.

Wt(W) E T Xt(w)M WE Q

then its covariant derivative along the Brownian paths is the vector field along

Cxt: 0 s t <~)given by

DWtlot=llt dldt(ll1-1w1) (47)

E. We shall not give the details but any Brownian motion on M from a point x0

can be considered as obtained in the way described: as the stochastic

development of a Brownian motion on IRn. To do this we have to anti-develop

our given Brownian motion on M. A simple way to do this is to express Xt as

the solution of some S.D.E. dxt = Y(xt) o dzt, as described in §3C of Chapter I.

Take a horizontal lift Y~ of Y to OM so Y~(u)e = Hu(Y(n(u))e) for (u,e) E OM x

IRm. Solve

(48)

from a given u0 E n-1(x0 ). This is a candidate for a horizontal lift of

Cxt: 0 s t <~)(assuming it does live as long as Xt, which it does [34], [43]).

For the solution set

T

Bt = J Ut -1(Y(Xt) o dzt) E IRn

0

(49)

One then has to use the martingale characterization of Brownian motion to

147311

show that Bt is a Brownian motion (or at least part of one). It is rather

straightforward to see that Xt: 0 => t <~is its stochastic development.

Note: we have used Y so that (48) and (49) can be used rather than

discussing equations like the stochastic versions of (12): however, even for

general semi-martingales z, the above construction gives an 'anti-development'

with resultant !An-valued process (Bt in our case) which is independent of the

choice of Y. In fact the horizontal lift (ut : 0 => t => ~ given by (48) is

independent of Y as is most easily seen by the uniqueness of solutions to the

local equations (12). The horizontality can most easily be expressed by

T

J &(odut)=O

0

(50)

for any stopping time 0 => T <~where for an lAP-valued 1-form e:TN-+ IRP on

some manifold N, and a continuous semimartingale (Yt : 0 => t < ~) in N we

define

T T

J S(o dyt) = J e(Y(yt) o dzt) (51)

0 0

where T is a stopping time less than ~ and dYt = Y(y1) o dzt for some Y and z.

This has to be shown to be independent of the choice of Y and z: but this is

easily done either by working with local expressions or by embedding N in

some IRq and extending e and any suitable Y over IRq.

Given the horizontal lift, parallel translation along sample paths, and

covariant differentiation along the paths can be defined by (46) and (47). The

integral (51) of a 1-form eon M along the paths of Xt can be written

T T

J e (o dxt) = J e(ut o dBt) (52)

0 0

Note that this anti-development can be carried out for semi-martingales

on M given any affine connection on M, as can the development itself: however a

Riemannian connection was needed to construct Brownian motions since we

needed the invariance of the distributions of Bt under the group O(n). See [70],

[76] for other situations.

148312

F. The classical Cartan development maps a smooth path

<5(t): 0 ~ t < oo on IRn (or T x M), starting at the origin, to a path in M starting 0

at x0 • Mathematically it is just as described above with Bt replaced by <S(t) in

( 43) to yield the determi ni sti c equation

du/dt = X(u(t)) (d6/dt) (53)

with u(O) = u0 a given frame at x0 • The resulting path x(t) on M, defined for all

time, it turns out, if M is complete, is that which is obtained by the classical

mechanical procedure of "placing M on a copy of IRn (by u0 ), then rolling M

along <5(t) : 0 ~ t < oo without slipping and taking x(t) to be the point of

contact of M with IRn at time t". The frame u(t) : IRn-+ T x(t)M represents the

way M is resting on IRn at time t. When trying to visualize, or sketch, the

situation it becomes clear that the natural objects to use are affine frames, as

in [64], rather than the linear frames which we have used.

§5. Examples: Spheres and hyperbolic spaces

A. A connected Riemannian manifold is orientable if its orthonormal frame

bundle OM has two components. This is always true for M simply connected.

An orientation is then the choice of one of these, to be called SOM: it will be a

principal SO(n)-bundle whose elements can be called oriented frames. For

example there is a natural choice when M = sn given by the inclusion of the

tangent spaces T xsn into 1Rn+1 and the natural orientation of !Rn+1.

B. The natural left action of SO(n+1) on 1Rn+1 restricts to one on sn. If N

= (1,0, ••• ,0) is the 'north pole' of sn (with abuse of geography) the map of

SO(n+1) to sn

p g -+ g.N (54)

induces a diffeomorphism of the quotient space (with a natural differential

structure which we do not need to examine)

J3: SO(n+1)/SO(n)-+ sn.

In fact we will show that the oriented frame bundle sosn can be identified

with SO(n+1). However first we need some remarks about SO(n+1) itself:

The Lie algebra ll(n+1) will be identified with the space of skew­

symmetric real matrices wlth .2Q(n) contained in it as

149313

~(n) = (g ~): 8 is n x n and B* = -B

There is then the vector space direct sum

s o(n+ 1) = so(n) + m

where

(0 -~t)

for s = ~ 0

with s written as a column.

Define

X : SO(n+ 1) x IRn -+ TSO(n+ 1)

by

(55)

(56)

so X(-)s is the left invariant vector field on SO(n+1) corresponding to the

element ! of m. Next define

by

ex: so(n+1)-+ sosn

ex(A)(s) = T A p(X(A)s)

= T 1(P o LA)!. (57)

To see this is an orthonormal frame it is only necessary to check that

T 1 (p o LA) : ill-+ T A.Nsn

is an isometry where m is given the inner product induced by s-+ f.. This is

easy to check when A is the identity, and follows for general A by noting that

SO(n+1) acts by isometries on sn (i.e. the derivative A*: T xsn-+ T A.xsn of x-+

Ax preserves the Riemannian metric for each x e sn). Similarly each ex(A) has

the correct orientation.

Finally observe that ex is equivariant for the right SO(n action, i.e. ex(Ag)

= ex(A)g:

In fact if Rg denotes right multiplication by g

poL A = (p o Rg -1) o LA = p o LA o Rg -1

forge SO(n), so

150

a(Ag)s = T 1 (p o LAg)t

= T1(P o LA Rg-1 Lg)t

= T 1(P o LA)(adg)-1t

= T 1 (p o LA)(gt)

314

(58)

(by an elementary computation). Thus p : SO(n+1)-+ sn has a principal bundle

structure isomorphic to 11: sosn-+ sn by (X.

There is a canonical connection on p: SO(n+1)-+ sn defined by

HA SO(n+1) = T 1 LA[m] A e SO(n+1) (59)

To see this is a connection note that forge SO(n)

T ARg[HA SO(n+1)] = T 1 LAg(adg)-1 [m]

= T 1 LAg(m) =HAg SO(n+1)

since m. is invariant under adg for g in SD(n). The connection form

w0 :TSO(n+1)-+ IRn is given by w0 (T 1LA(a + t)) =a for A e SO(n+1), a E ll(n)

and t em.

An important property is that it is invariant under the automorphism o0 of

SO(n+1) given by cs 0 = SAS where S = (-6 ~)

This connection induces a connection on n : sosn -+ sn via ex:, i.e. a

Riemannian connection on sn. In fact this is the Levi-Civita connection, a fact

which is proved using the invariance under cs 0 just mentioned: see [64] page

303.

C. This situation works much more generally. Another important example is

obtained by taking the lorentz group 0(1,n) of linear transformations of 1Rn+1

which preserve the quadratic form (Sx,x) for S as in §B above. This has 4

components: let G be its identity component,

G =A e 0(1,n): det A= 1 and A11 2: 1). (60)

Then the Lie algebra !J_ of G is just .Q. (n,1) where

Q(n,1) =A e O..(IRn+1;1Rn+1):Ats + stA = O. (61)

Let

M = x e 1Rn+1: (Sx,x> = -1 and x1 2: 1. (62)

Then the form v-+ (Sv,v) induces a Riemannian metric on M by restricting it to

each T xM and G acts on the left on M, preserving this structure with

151315

p:G-+M

given by p(A) = A.N for N = (1,0, ••• ,0), inducing a diffeomorphism of G/H with M

for

H =A e G : A.N = N = (~ ~):BE SO(n).

We can identify H with SO(n).

As before there is commutative diagram

a G SOM

~/ M

with ex equivariant under the right action of SO(n), so that p can be identified

with the oriented frame bundle n. For G there is the splitting

.Q.(1,n) = so(n) + m*

for

m* = (t* : s e !Rn, where t* = (~ ~J (63)

Again there is a connection ro 0 which induces the Levi-Civita connection for M,

see [64], p. 303, and an involution cs 0 •

The manifold M is n-dtmensional hyperbolic space Hn , see Exercise (ii)

at the end of §8 below and §4G of Chapter III.

The general situation where this works is that of a Riemannian

symmetric space see [64], [65].

D. In these situations the stochastic development construction is equivalent

to solving dgt = Xgt) o dBt for the X defined on G, (e.g. on SO(n+1) for sn), and

projecting by p to get Brownian motion on M. Thus for our solution with g0 = 1,

gt • N : t 2:: 0 is a Brownian motion on M starting at N.

Remark:

It is shown in [43] p. 257 that when M is a Riemannian symmetric space,

152316

so we can identify M with G/H as above, Brownian motions on G themselves

project to Brownian motions on M, so if g1 : t 2: o is a S.M. on G from 1 then

(gt.N:t 2: N is one on M starting at N, where N eM is arbitrary.

Similarly, for example by the discussion of the S(t,u)e in the proof of the

Ito formula, Theorem 4C, the geodesics of M from a point N are given by

((ExptA).N: t e !R where Exp is the exponential map of the group G, see §1A,

and A lies in a certain n-dimensional subspace of the Lie algebra of G : in

particular for M = sn we need only take A em, and for M = Hn we can take

A Em*.

§6. Left 1nvar1ant S.O.S. on Lte groups

A. For G a Lie group with left invariant Riemannian metric (i.e. each Lg is an

isometry) there is the left invariant system

dgt = X(gt) o dBt (64)

where n = m and X(1): !Rn-+ T 1 G is some isometry and X(g)e = Lg(X(1)e) fore e

!Rn and g E G. The solutions will be Brownian motions if L: VXP(XP(x)) = 0; p

in particular if each VXP(XP(x)) = 0. The latter means precisely that the

integral curves of d~/dt = XP(~t) are geodesics (just differentiate this equation

along its solution). For a Lie group with both left and right invariant metric

the i nversi ng map g-+ g -1 is an isometry so that if

)'(t): t E !R) is a geodesic so is y(t)-1: t E !R, and therefore )'(t)-1 = )'(-t). From this one can deduce that the geodesics are precisely the one-parameter

groups, i.e. the solutions of cr(t) = X(<l(t))e for some e (e.g. see [79]). Thus in

this bi-invariant case the solutions to (64) are Brownian motions.

The compact groups admit bi-invariant metrics, and conversely every G

with a bi-invariant metric is a product G' x !Rk with G' compact.

B. Given a bi-invariant metric on G we have just persuaded ourselves that

VA(A(x)) = 0 for all left invariant vector fields. Therefore if 8 and C are both

left invariant, by taking A = 8 + C we obtain, using (24),

VB(C(x)) = Hc,B](x) (65)

§7. The Second Fundamental form and gradient S.O.S. for an embedded

submani fold.

A. Suppose now that M is a submanifold of !Rn with induced Riemannian

metric. There is then a natural S.D.S. (X,B) on M where Bt: t 2: 0 is Brownian

153317

motion on IRm and X is just the orthogonal projection map P of §10, Chapter I,

(as in the S.O.E. of equation (7)) so X(x) : IRm -+ T xM is the orthogonal

projection.

Suppose f: M-+ IRis c1. Let f 0 : IRm-+ IR be some smooth extension. Using

V 0 for the gradient operator on functions on IRm we have df(v) = df 0 (v) for v e

T xM and so Vf(x) = X(x)(V 0 f 0 (x)) for x e M. Thus if <p : M -+ IR m denotes the

inclusion, writing <p(x) = (<p1(x), ••• ,<pm(x)) we see

XP(x) = VtpP(x) x e M,p = 1, ••• ,m. (66)

For this reason our S.O.S. is often called the gradient Brownian system for

the submanifold (or for the embedding <p). We will show that its solutions are

Brownian motions on M. For this we need ! VXP(XP(x)) = 0 all x eM, and so p

we will first examine how the covariant derivative for M is related to

differentiation in IRm.

B. Suppose Z is a vector field on M. Take some smooth extension which we

will write z0 : IRm -+ IRm. For vxM = (T xM)~ in IRm, as in §10 of Chapter I,

there is a symmetric billnear map

cx:x : T xM x T xM-+ vxM

called the second fundamental form of M at x , such that Gauss's formula

holds: for veT xM

OZ 0 (x)(v) = VZ(v) + CX:x(Z(x),v) (67)

One way to prove this is to define VZ(v) to be the tangential component of

OZ 0 (x)(v) and write Vx(Z 0 ,v) for its normal component. Then one can verify

that (Z,v) -+ VZ(v) satisfy the conditions which ensure it is the covariant

differentiation operator for the Levi-Civita connection on M and furthermore

show that Vx(Z 0 ,v) has the given form for a symmetric CX:x, e.g. see [65], pp.

10-13.

From this there is the bilinear map for each x in M:

Ax : T xM x VxM -+ T xM

defined by

<Ax(u,n,v> = <cx:x(u,v),S). (68)

If s:M-+ IRm is c1 with s(x) E VxM for all X E M and So is a c1 extension then

Weingarten's formula gives

154318

Ds 0 (x)(v) = -Ax(v,s(x)) +a normal component

In fact for x EM

(Z0 (x), s0 (x)) = 0

for Z and Z0 as before. Therefore if vET xM

(DZ0 (x)v, s0 (x)) + (Z0 (x), Ds 0 (x)v) = 0

i.e.

(69)

< o::x(Z 0 (x),v), s 0 (x)) + <Z 0 (x), tangential component of Ds 0 (x)v) = 0 proving

(69).

C. The following goes back to Ito's work published in 1950:

Proposltion 7C. The solutions of the gradient Brownian system for a

submanifold M of IRm are Brownian motions on M.

Proof For the constant vector fields EP(x) = (o1P, ••. ,omP), p = 1 tom on IRm,

EP(x) = X(x)EP(x) + Q(x)EP(x) for Q(x) = 1d- X(x). Therefore differentiating and

taking the tangential component, for veT xM

o = vxP(v)- Ax(v,Q(x)EP(x)). (70)

Thus if we choose our orthonormal base e1, ... ,em of IRm so that e1, ... ,en

are tangent to M at x

vxP(v) = o p = 1 to n (71)

while

vxP(xP(x)) = o p = n+1, ••. ,m

because XP(x) = 0 for such p.

Thus

k VXP(XP(x)) = 0 p

as required. I I

Note that if X0 : IRm-+ IL(IRm,IRm) extends X then the equation

dxt = X0 (Xt) o dBt

whose solutions lie on M when starting on M, has Ito form

dxt = X0 (xt)dBt + ~ k DX0 P(xt)(X 0 P)(xt))dt p

and for x EM

k DX0 P(x)(X 0 P(x)) = L O::x(X0 P(x), X0 P(x)) = trace O::x p p

by (72) and (67). The standard example of this is the equation

(71)'

(72)

155319

dxt = dBt- <xt,dBt> lxtl-2xt- Hm-1)lxtr2 x1dt (73)

which gives Brownian motion on the sphere sm-1(r) of radius r if

x0 E sm-1(r). For variations, extensions, and further references consult

[95].

Note that from (70)

di v XP(x) == trace Ax( -,Q(x)EP(x)) L~= 1 ( a:x(Eq(x), Eq(x) ), Q(x)EP(x))

= L~=l (trace a:x, Q(x)EP(x)) (74)

D. The mean curvature normal at x is 1/n trace a:x E VxM. For a

hypersurface vxM is a 1-dimensional subspace of IRm and a choice of

orientation in vM (e.g. outward normal) gives the mean curvature as a real

valued function on M.

Similarly for a hypersurface the second fundamental form can be treated

as a real valued bilinear form on the tangent spaces to M. At a point x its

eigenvalues are called the principal curvatures at x, and its eigenfunctions

are the principal directions at x.

§8. Curvature and the derivat1ve flow

A. Given intervals I and J of IR and a piecewise c1 map u: I x J-+ M, there are

vector fields ou/os and ou/ot over U (i.e. the derivatives with respect to the

first and second variables respectively). Taking normal coordinates centred

at a point u(s,t) it is immediate from (17) that

D/ot au;as = D/os au;at (75)

If F t : M x Q-+ M, t 2: 0, is a smooth flow for our S.O.S. and v E T xM, choose

o: [-1,1]-+ IR with 6(0) = x and cr(O) = v. Then

T xFt(v) = a;as Ft(CI(s)) a.s. (76)

Defining parallel translation via a horizontal lift as in §§4D, E, we can

covariantly differentiate (76) in t using the analogue of (47) as definition, to

get

Dvt = Do/os Ft(CI(s))

for Vt = T x F1(v) : Q-+ T x1M, and so by the analogue of (75), proved in exactly

the same way,

Dvt = 0/oS Y(Ft(CIs)) • dZt

156320

i.e.

(77)

For later use, and as an exercise, we will find an equation for lv1F assuming now that we have a Riemannian metric and are using its Levi-Civita

connection. Certainly there is the Stratonovich equation

dlvtl2 = 2(vt, o Dvt>xt = 2<vt, VY(vt) o dzt>xt (78)

obtained, for example, by parallel translation back to x. A safe way to get the

Ito form is to use the Ito formulae (5): let S(t,x)(e) denote the flow of Y(-)(e)

and set &S(t,v) = T x(S(t,x)(e))(v) for veT xM, so

D/ot &S(t,v) = VY(&S(t,v))(e).

Therefore

d/dt !&S(t,v)l2 = 2(&S(t,v), VY(&S(t,v))e>

= 2(v, VY(v)e> at t = o and, at t = 0,

d2/dt2 I &S(t,v)j2 = 2(VY(v)e, VY(v)e) + 2(v,VY(VY(v)e)e)

+ 2(v,v2(Y(x)e,v)e) (79)

where, for Z a vector field and u e T xM,

v2z(u,-) = v u(VZ) : T xM + T xM (80)

Thus

dlvtl2 = 2(Vt,VY(vt)dzt> + (VY(vt)dZt,VY(vt)dzt>

+ <vt, VY(VY(vt)dzt)dzt) + <vt,v2Y(Y(x1)dzt,vt)dzt> (81)

(To be convinced of the applicability of equation (5) observe that Vt is actually

a solution of the S.D.E. on TM, dvt = &Y(vt) o dzt, given in §4E, Chapter I.) We

shall simplify (81) in §8C below.

B. The curvature R of an affine connection is a section of the bundle IL(TM,

TM; IL(TM;TM)) so it can be considered as a map

R : TM ffi TM -+ IL(TM;TM)

where TM ffi TM = U(T xM ffi T xM : x e M), such that

u,v,w-+ R(u,v)w

is tri-linear in u,v,w e T xM· It can be defined by

R(u,v)w = v2w(u,v) - v2W(v,u) (82)

where W is a vector field such that W(x) = w at the given point x of M. We will

157321

see below in §9C that this definition gives a result independent of the choice of

such W. If U, V, Ware all vector fields there is the new one R(U,V)W given by

x -+ R(U(x), V(x))W(x)

so

R(U,V)W = Vu Vv W- Vv VuW- V[u,v]W (83)

for a torsion free connection using (24) and §1D, Chapter II).

From 82) R is anti-symmetric in its first two variables. For the Levi­

Civita connection of a Riemannian metric it turns out that (R(u1,v1)u2,v2>x is

antisymmetric in u1, v1 and in u2, v2 and satisfies

(R(u 1,v1)u2,v2>x = (R(u2,v2)u1,v1>x 84)

This is an automatic consequence of the skew-symmetry and the relation

R(u,v)w + R(v,w)u + R(w,u)v = 0 (85)

which can be proved by choosing vector fields U, V, W with U(x) = u, V(x) = v,

W(x) =wand which commute i.e. [U,V] = [V,W] = [W,U] = 0. For details see e.g.

[64] or [79] (where a different sign is used!).

For the Levi-Civita connection the Riemannian curvature tensor is defined

by R(v1, v2, v3, v4) = (R(v1, v2)v4, v3>x

=- (R(v1,v2)v3,v4>x (86).

for vi e T xM, (note the sign difference from [33]) and the sectional curvature

Kp(x) of a plane P in T xM is

Kp(x) = R(vv v2, v1, v2) (87)

where v 1, v2 is an orthonormal base for P.

When M is a surface, i.e. n = 2, there is a unique P at x, namely P = T xM,

and then Kp(x) coincides with the classical Gaussian curvature of M at x. For

M a submanifold of IR3 wHh induced metric it was Gauss's famous theorem

aegregium which showed

Kpx) = :>.. 1 :>..2 (87)

where :>..v :>.. 2 are the principal curvatures at x §7D), see [65] for example.

Continuing with the Riemannian case there is the Ricci curvature

Ric : TN EB TN -+ IR

Ric(v 1,v2) =trace [v-+ Rv,v 1)v2J = L. (R(ei,v 1)v2,ei>

J

(88)

for e1, ... ,en an o.n. base in T xM· Clearly it is symmetric. Observe that Ric(v,v)

158322

is the sum of the sectional curvatures of any family of (n-1) mutually

orthogonal planes P in T xM containing v, provided lvl = 1. Thus bounds on the

Ricci curvature are weaker assumptions than bounds on the sectional

curvatures.

The Ricci curvature plays a very important role in the study of diffusions.

An important result of differential geometry is that if the Ricci curvature is

bounded below i.e. if there exists a constant C with

Ric(u,u) 2: Clul2 for all u e TM

then for any x0 E M if r(x) = d(x,x 0 ) then t::,r is bounded above uniformly at

all points where r is differentiable [101], [58]. From the Ito formula

applied tor as if it were c2 the result of 5.-T. Yau that a complete

Riemannian manifold with Ricci curvature bounded below is stochastically

complete (i.e. its Brownian motion does not explode), is no surprise [43], p.

242: see [38] for an analytic proof, and [ 61] for a probabilistic version with

an Ito formula for r(xt) involving a local time at the points where it is not

differentiable.

Taking the trace of the Ricci curvature at x gives the scalar curvature, a

real valued function K : M-+ IR

Note that for a surface K(x) is twice the Gaussian curvature

K(x) = Ric(e 1,e 1) + Ric(e2 ,e 2) = 2Ric(eve1)

(89)

= 2Kp(X) (90)

A space is said to have constant curvature if all its sectional curvatures

Kp(x) are the same (it suffices to know they are independent of P for each

x EM by a result of Schur, see [64]). If so

R(u,v)w = k ((w,v)u- (w,u)v) (91)

where k = Kp(x). For such a space of constant curvature k we see

Ric(u,v) = (n-1)k (u,v>x (92)

and

K(x) = n(n-1)k (93)

C. We can now get improved formulae for the norms of the derivative flow.

Suppose the original S.D.E. was dxt = X(x1) o dBt + A(xt)dt with generator A=

159323

~t. + Z, where Z is a vector field and M is Riemannian. Write A= W + z. Then,

for each x e M, by §4A

~ L VXP(XP(x)) + W(x) = 0 p

Differentiating this in the direction of v, for veT x M

Lp v2 XP(v,XP(x)) + Lp VXP(VXP(v)) + 2VW(v) = 0

whence, by (82) and the definition (88) of Ric

2: (V2XP(XP(x),v),v) + Ric(v,v) + (2: VXP(VXP(v)) + 2VW(v),v) = 0 (94) p p

Substituting in (81):

dlv1!2 = 2(Vt,VX(v1)dBt) + 2(Vt,VZ(Vt))dt

- Ric(vt,Vt)dt + :l:p (VXP(v1),VXP(vt))dt (95)

In particular we see lv11 e L2(Q,~,IP) provided the quadratic forms

v-+ 2(v,VZ(v)) -Ric (v,v) and

v-+ IVXP(v)l2, p = 1,2, ••• ,m

for v e T xM are bounded above uniformly over M. Equation (95) and the growth

of lvtl will be examined in detail in special cases in Chapter III.

§9. Curvature and torsion forms

A. When trying to find a useful analogue of (~lS) for the canonical S.D.S. on

the frame bundle of a Riemannian manifold we shall need the curvature form,

so we will describe it here and use it to prove some of the basic results about

curvature which were stated in §8.

First suppose we have a principal G-bundle n : B -+ M with a connection

form & • This is a g-valued 1-form on B. As with a real valued one-form it

has exterior derivative

d& : TB ffi TB -+ g

(v 1,v2)-+ d&(v 1,v2)

which is antisymmetric, satisfying

dw(U(b),V(b)) = U(&(V))(b)- V(&(U))(b)- &([U,V](b)) (96)

for U,V vector fields, where &(V) is the g-valued function on B, b -+ &(V(b)),

etc; see the discussion of differential forms in Chapter IV.

N.B. Here we are departing from the convention of Kobayashi and Nomizu

160324

[64], which would have a factor of~ multiplying the right hand side of (96).

The curvature form is the 2-form R given by

R : TB EB TB -+ g

(97)

for v1, v2 e TbB where h : TB -+ HTB is the projection onto the horizontal

subspace. By the invariance property (B) of w, i.e. w o TRg = ad(g-1) ow , we

have

(98)

for b e 8, g e G, and v1 , v2 e TbB, because of the invariance of exterior

differentiation under diffeomorphisms: in our case

d(ro o TRg)(v 1,v2) = dro(TRg(v 1),TRg(v2)).

B. To see the importance of R suppose v1, v2 are horizontal vector fields on

B so Vi(b) e HTbB for each b. Then ro(Vi(b)) = o for all band so by (96):

R(V 1(b), V2(b)) =- ro([V 1,V2 ](b)) (99)

Thus if R. vanishes identically the Lie bracket of horizontal vectors is

horizontal: but this is precisely the classical necessary and sufficient

condition of Frobenius' theorem for the integrability of (HTbB: be B) i.e. for

the existence of a submanifold through each b of B with HTbB as its tangent

space. (In particular if this happens for a Levi-Civita connection the canonical

S.D.S. will be far from hypo-elliptic).

C. Now suppose we have an affine connection, so B = GLM, or OM for the

Riemannian case. Then there is the canonical1-form

e : TB -+ !Rn

e(v) = u-1(TTT(v)) veT uB

From this we obtain the torsion form

®(v 1,v2) = d9(hv 1,hv2)

It satisfies

8(TRgv1, TRgv2) = g-1 ®(v1,v2)

Because of (98) and (102) there exist

T : TM EB TM -+ TM

bilinear and skew symmetric from T xM EB T xM-+ T xM and

R : TM EB TM -+ IL(TM;TM)

again bilinear and skew symmetric, defined by

(100)

(101)

(102)

161325

(103)

and

R(v1,v2)v3 = u R (Huv1, Huv2)u-1(v3) (104)

for v; e T xM, u e n-1(x), and Hu the horizontal lift operator.

These are the torsion and curvature tensors of our connection. We must

show that this definition of R coincides with the one in §8:

Proposition 9C.

ForT, R defined by (104), if V,W are vector fields on M and v1 ,v 2

tangent vectors at x toM

( i) If T = 0 then

v2v(v1,v2)- v2v(v2,v1) = R(v1,v2)V(x)

(ii) [V,W](x) = VW(V(x))- VV(W(x))- T(V(x),W(x)) (104a)

In particular the connection is torsion free iff T = 0 or equivalently® = 0.

~

(i) Choose u0 e n-1(x) and define the vector field V; on 8 by

V;(u) = Hu(uu0 -1v;) i = 1,2 ..

If T = o then

0 = ®(V 1,v2) = d9(V 1,V2)

= v 1 ecv2)-v2e(v 1) - e([v 1,v2])

= - e([v 1.V2D

since e(V;) is constant for each i. Thus Tn([VvV 2 ]) = o and so [v 1,v 2 ] is

vertical. It follows that

[V 1,v2](u0 ) = A*(u0 )

for

A= ro([V1,V2](u 0 )) =- R(V1(x), V2(x))

(by equation (9) and (99)).

Now by the definitlon, (19), for v(u) = u-1v(n(u)), etc.

V2V(V 1•V2) = u0 d(Vv)(V 1 (u0 ))(u0 -1v2)

= Uo V1(Vv(-)(ua-1v2))

= u0 V 1V2(V)(u0 ) (105)

Thus V2V(v1,v2)- v2v(v2,v1) = u0 [V 1•V2JV(uo)

= u0 A* v(u0 )

162

as required.

326

= u0 d/dt V(u0 exptA) I t=O

= u0 d/dt exp(- tA)u 0 -1v(x) I t=O

= -u0 Au0 -1 V(x)

= u0 R(v1,v 2)u 0 -1v(x)

Part (ii) can be proved similarly. II

D. Example

Let us compute the curvature of the sphere sn using the identification of

sosn with SO(n+1) as in §5. Using the notation of §5 to computeR., because of

left invariance it is enough to compute it at the point 1. For this let S• 11 E !Rn

with corresponding t and n. in m.. Take the left invariant vector fields U, V on

SO(n+ 1) which are t and n. at 1. Then

[U,V](1) = [t, .nJ =a e ~(n) which confirms that [U,V] is vertical (cf. the proof of Proposition 9C).

Therefore, by definition of the connection form ro 0 and equation (99), if Q is

the curvature form

R(t, n.) = - .e.. Now if v E !Rn, (identified with (O,v) in 1Rn+1) we see

rt.n.Jv = -<1l.v>s + <s.v>'Tl

Also T 1 TI: T 1 SO(n+1)-+ TNsn c!Rn+1 is just

A -+ ( O,cx:)

(106)

where ex: is the component of A in m., since TI maps g to g.N. Thus, if vi e TNSn

= 0) x !Rn c 1Rn+1, fori = 1,2,3, the horizontal lift H1(vi) = Y.i and

R(v1,v2)v3 = R(Y.i,Y.i)v3

= <v2,v3)v 1 - <v 1•v3>v2

which shows that sn has constant curvature+ 1, (c.f. equation (91)).

Exercises

(i) Check that the torsion form ® vanishes identically, so that we have a

complete proof that ro 0 gives the Levi-Civita connection.

(ii) Do the same for hyperbolic space Hn, showing it has constant curvature -

1; (the difference is in the use oft* given by (63) instead of tl

163327

110. The derivat1ve of the canonical flow

A. Let M be a Riemannian manifold with Levi-Civita connection. Suppose, for

simplicity of notation, that its canonical S.O.S. (X,B) on OM is strongly complete

with flow Ft(u,w): t 2:: 0, u e OM, we Q). To get equations like those of §SA for

TuFt we would need an affine connection for OM rather than just M. There are

various candidates, e.g. see [77], but the computations get rather complicated

and we will go by a direct method as in [43] (which in fact boils down to using

the 'canonical flat connection' for OM).

The derivative flow lives on TOM. However there is a canonical

trivializatlon of this tangent bundle: i.e. a diffeomorphism

\j) : TOM -+ M x !Rn x Q(n)

which restricts to a linear isomorphism \j)u: T uOM-+ u x IRn x Q(n) for each

u e OM. This is given for V e T uOM by

\j)(V) = (u,9(V), ro(V)) (107)

fore, ro the fundamental form and connection form. Using this, for V0 e T u OM 0

we will describe TF 1(V 0 ) in terms of !Rn- and Q.(n)-valued processes st =

9(TFt(V 0 )), At= ro(TFt(V 0 )). This is similar to a procedure used in the

Malliavin calculus [17], (although here we have differentiation with respect to

the initial point, and there a 'differentiation' with respect to the basic noise

Bt: t 2:: 0 is being considered as in [42a]) ••

We will use the analogous notation to §SA so S(t,u)e gives the flow on OM

of ut = X(u1)(e) fore e !Rn, and &S(t,V) gives its derivative flow on TOM.

B. A vector field J along a geodesic 1 in M is a Jacobi field if it satisfies

o2J;at2 + R(J, dl/dt) d"t/dt = o (10S)

They arise as infinitesimal variations of geodesics, [79], (see the proof

below) and as a method of computing derivatives of the exponential map.

Lemma 108. For V e T uOM and fixed e in !Rn set J(t,V) = Tn(&S(t,V)). Then

J( -, V) is a Jacobi field with

J(O,V) = Tn(V)

D/ot J(t, V)lt=O = u ro(V)(e). (109)

Proof First recall from §4C, equation (44), that t-+ n(S(t,u)e) = lt(u,e) is a

164328

geodesic: so J(-,V) is a vector field along a geodesic. Next take a horizontal

path o in OM with o(O) = u and 0"(0) = hV, and the path gin O(n) given by

g(s) = exp s ro(V).

Set p(s) = o(s).g(s). Then p(O) = V

so that

J(t,V) = olos (nS(t,p(s))e)]s=O

= olos 6't(p(s),e)ls=o·

Therefore

Dlot J(t,V) = Dlos a1at lt(p(s),e)ls=O

and by (82)

o21at2 J(t,V) = Dlos Dlot 'Yt(p(s)),e)ls=O

-R(J(t,V), y1(u,e)) y1(u,e)

(110)

which shows J is a Jacobi field since Dlot olot lt(p(s),e) = o because l is a

geodesic.

Clearly J(O,V) = Tn(V) as claimed, and also by (110) and the proof of

Theorem 4C:

Dlot Jlt=O = Dlos p(s)els=O = Dlos o(s).g(s)els=O

= o(s) dlds g(s)els=O = u ro(V)e. 1 I

C. From the lemma we have

dldt e(oS(t,V)) = dldt (S(t,u)e)-1 Tn(oS(t,V))

= (S(t,u)e)-1 Dlot J(t,V)

= ro(V)e

at t = 0, and, at t = 0,

ct21dt2 e(oS(t,V)) = u-1 o21at2 J(t,V)

= -u-1 R(TnV,ue)ue

(111)

(112)

Now take any affine connection for the manifold OM which is torsion free

(e.g. the Levi-connection for the metric on OM induced by t.p ). For this

dldt ro (oS(t,V)) = Vro(X(S(t,u)e)e)(oS(t,V)) + ro(DiotoS(t,V)) (113)

However, by §SA,

Dlot oS(t,V) = VX(oS(t,V))e

and also since X(u)e is horizontal, ro(X(u)e) = 0 for all u e OM whence

Vro(V)(X(u)e) + ro(VX(V)e) = 0 (114)

165329

for all VeT uOM. Substituting in (113)

dldt &(&S(t,V)) = V&(X(S(t,u)e)e)(&S(t,V))

- V ro(&S(t, V))(X(S(t,u)e)e)

= d&(X(S(t,u)e)e, &S(t,V))

by (96) (choosing suitable U,V which commute at S(t,u)e).

(115)

To proceed further we need the first of the following, and we state the

second in passing; they are valld for any affine connection, and the first for

any connection on a principal bundle:

Structure Equations: For V1,V2 e T uOM

(i) d&(V 1,v2) =- [&(V 1), &(V2]) + R(V 1,V2)

(ii) d8(V 1,V2) =- (&(V 1)e(V2)- &(V2)e(V 1)) + 0(V 1,v2) (117)

Proof We prove only (i), for (ii) see [64] Theorem 2.4, Chapter III, (the ~·s in

[64] come from the different convention for exterior differentiation used

there).

For (i): if both V 1, v2 are horizontal the result is clear by definition of R.

If both are vertical we can suppose V 1 = A *(u), v2 = B*(u) at the given point u,

for A, B e Q(n). Then

d&(A*,B*) = A*&(B*)- B*ro(A*)- ro([A*,B*J)

= - [A,B] = - [w(A *), w(B*)]

since ro(B*) and w(A*) are constant and [A*,B*] = [A,B]*. This gives (i) in

this case because R(A*,B*) = 0.

To complete the proof it is enough to suppose v1 is horizontal, v1 = V(u)

for some horizontal vector field V, say, and v2 "' B*(u) some Be Q(n). Again

d&(V 1,V2) = V ro(B*(u))- B*w(V 1)- ro([V,B*](u))

= - ro([V ,B*](u)).

Thus (i) is equivalent to [V,B*] being horizontal. However this is clear since

[V,B*](u) = dldt TWt(V(u.exp(-tB)))!t=O where Wtu = u.exp(tB) and horizontality

is preserved under right translation. I I

Applying the first structure equation to ( 115):

dldt &(&S(t,V)) = R(X(S(t,u)e)e, &S(t,V)) (118)

= (S(t,u)e)-1R((S(t,u)e)e, J(t,V))S(t,u)e (119)

and so, at t = 0

166330

d2/dt2 ro(&S(t,V)) = u-1 D/ot R((S(t,u)e)e, J(t,V))It=O u

= u-1 VR(ue)(ue, ue(V))u

+ u-1 R(ue, u ro(V)e)u

since D/t (S(t,u)e)e = 0.

(120)

From these we have our equations, in Stratonovich form by (111) and

(119):

dst = At o dBt

dAt = Ut-1 R(Ut 0 dBt, Ut St)Ut

(121a)

(121b)

and in Ito form using (112) and (120), for e1, .•• ,en an orthonormal base for IRn:

dst =At dBt- ~ Ut-1 Ric(ut svt (122a)

dAt = u1-1 R(ut dBt,Utst)Ut + ~ (ut-1 Z:i VR(utei)(Utei,utst)Ut

(122b)

(It is shown in [43] p. 168 that the covariant derivative of R can be replaced by

a term in the covariant derivative of the Ricci tensor since for an orthonormal

base f 1, ... ,f n for T xM

((Z:i VR(fi)(fi,w))v 1,v 2>x = VRic(v2)(v 1,w)- VRic(v 1)(v 2,w) (123)

for all v1,v2, win T xM.)

Note that if dBt is replaced by dt in (121a,b) we obtain the Jacobi field

equation for ut St· The fields Tn(TF1(V)): t 2:: 0 along the Brownian motion

n Ft(u): t 2:: 0) have been called stochastic Jacobi fields, [71].

167331

CHAPTER III: CHARACTERISTIC EXPONENTS FOR STOCHASTIC FLOWS

§.1. The Lyapunov Spectrum

A. Suppose throughout this section that M is a compact connected Riemannian

manifold with smooth S.D.S.

dx1 = X(xt) o dBt + A(xt)dt

differential generator A. Let Ft: M x Q-+ M, t;:: 0 denote its flow so for each

w E Q we have a C00 diffeomorphism Ft(-): M-+ M, derivative TFt(-,w) : TM-+

TM. Let (Q,~,!P) be the classical Wiener space of paths starting at 0 in lRm

with Bt(w) = w(t), and let 9t: Q-+ Q be the shift:

9t(w)(s) = w(t+s)-w(t) (124)

Then lP is invariant under e1 fort;:: 0.

A Borel probability measure p on M is invariant for our S.D.S. if

IE p o Ft(-,w)-1 = p t;:: 0 (125)

Since M is compact there exists an invariant measure (e.g. see [102], XIII

§4). The invariance of p depends only on A, not on the choice of S.O.S. with .A.

as generator. When A is elliptic then pis a smooth measure i.e. p(dx) = :>..(x)dx

for some smooth A. where dx refers to the Riemannian volume element: it is

also unique. This is because :>.. is a solution to the adjoint operator equation

(e.g. see [59]).

Define

'Pt : M X Q -+ M X Q

by

'Pt(x,w) = CFt(x,w), e1w).

Then for each s,t;:: 0

'Pt 'Ps = 'Pt+s a.s.

since

Ft<Fs(x,w), esw) = Fs+t(x,w) a.s.

Also if p is invariant for the S.D.S., then p 0 lP is invariant for 'Pt since if

f: M x Q-+ lR is integrable

168332

Jf f(Ft(x,w),e1w)p(dx) IP(dw)

= f Jf f(Ft(X,W1), 8tW2)p(dx) 1P(dw1)1P(dw2)

(because e1w is independent of Ft(x,w))

= If f(x,w) p(dx) IP(dw).

Say that p is ergodic if p 0 IP is ergodic for ~t : t 2: 0 i.e. if the only

measurable sets in M x Q which are invariant under ~ 1 : t 2: 0 have p 0 IP­

measure 1 or 0. This agrees with the definition in [102]. An ergodic

decomposition for any invariant pis given in [102].

B. In this chapter we shall be mainly concerned with looking at special

examples of the following version by Carverhi1l [20] of Ruelle's ergodic theory

of dynamical systems:

Theorem 18 Let p be an invariant probability measure for A • Then there

is a set r c M x Q of full p 0 IP-measure such that for each

(x,w) E r there exist numbers

:>-_(r)(x) < .•• < :>-_(1)(x)

and an associated filtration by linear subspaces ofT xM

0 = v(r+1)(x,w) c v(r)(x,w) c ... c v(1)(x,w) = T xM

such that if

then

V E V(j) (X,W) - V(j+1)(X,W)

lim 1/t log IITFt(v,w)IJ = :>-_(j)(x)

t-+oo

(126)

where II II denotes the norm ustng the Riemannian metric of M. Moreover

for (x,w) E r the multiplicities mi(x) := dim v<Dcx,w) - dim v(i+1)(x,w) do not

depend on w and if

(127)

then

:>-.L:(x) =lim 1/t log det T xFt(-,w). (128)

(strictly speaking we should write ldet T xFt(-,w)l here and below or use some

other convention to ensure it is continuous in t with value 1 at t = 0) •

Proof

Following [20] embed M in IRn+p for some p and extend X,A just as in §2C of

169333

Chapter I to give X, A on IRn+p with compact support. Let Ft(x,w) refer to the

flow of this system, and <l)t to the flow on !Rn+p x Q; p remains an invariant

measure on IRn+p, concentrated on M.

Fix some time T > 0 and for n = 0,1,2, ••• set

Gn(x,w) = DFT(<l>nT(x,w)): IRn+p-+ !Rn+p

(where the D refers to differentiation in IRn+P). Set

Gn(x,w) = Gn_ 1(x,w) o ••• o G0 (x,w)

so

Gn(x,w) = DFnT(x,w),

by the chain rule.

and

Fairly standard estimates show that both

log+ IIDFT(x,w)ll e L 1(M x Q; p ® IP) (129)

(130)

We can therefore apply the Oseledec multiplicative ergodic theorem, as in

00 Ruelle [87], to (Gn(x,w) n=O and obtain the theorem in a discrete time version,

for X, A, and with the :Ai and mi possibly depending on (x,w).

To deduce the continuous time version from this as in [87] the

integrability of sup log IIDF t(x,w )II and of

O~t~T

sup log I!D(FT(-,w) o F1(-,w)-1)Ft(x,w)ll

O~t~T

are used. Having done this the filtrations for Ft on M are obtained by

intersecting those for F with each T xM. To show that the :Ai and mj do not

depend on w as described, [24], use the fact that this is certainly true if p is

ergodic (in which case they can be taken independent of x for suitable choice of

r), and then the fact that, even if not ergodic, p can be decomposed into ergodic

measures concentrated on disjoint subsets of M, [102]. I I

C. When the system is non-degenerate p is ergodic, and unique, and r can be

170334

chosen so that the exponents Ai and multiplicities mi, and consequently the

mean exponent 11n AL;, are all independent of x.

D. When p is ergodic and an exponent Ai is negative there is a stable

manifold theorem due to Carverhill [20]. Its proof comes from Ruelle's in

[87] for deterministic systems using the embedding method and regularity

estimate, for T > 0,

IE sup IIF t (-, w) II 2

< oo o:;:;~ c

Theorem 10 (Stable manifold theorem) For ergodic p with A(j) < 0 for

some j the set r of Theorem 18 can be chosen so that for each (x,w) E r the set y(j)(x,w) given by

y(j)(x,w) = y: lim sup 11t log d(F t<x.w ),F 1(y,w)) ~ A (j)) ( 131) t~ 00

is the image of v(i)(x,w) under a smooth immersion tangent to the identity at

X. I I

This means there is a smooth f : v(i)(x,w) ~ M which has T vf injective for

each v in v(i)(x,w) and has T 0 f = 1d, with yj)(x,w) as its image. It will be

locally a diffeomorphism onto its image but not necessarily globally.

If A (1

) < 0 we say the system is stable. In this case y(1)(x,w) will be an

open subset of M.

N.B. When M is compact any two Riemannian metrics ((-,->x: x EM

(-,->'x: x EM are equivalent: there exists c > 0 with c-1(u,u>'x ~ (u,u>x ~

c(u,u)'x for u e: TxM· Consequently the Lyapunov spectrum and stable

manifolds are independent of the choice of metrics. This would not be true if

we allowed M to be non-compact e.g. as in [26], [27).

E. The first examples to be studied were the 'noisy North-South flow on s1·

(i.e. in stereographic projection so the North pole corresponds to oo and the

South pole to 0 the equation is

dXt = e: Y(Xt) o dBt - dt

where (Bt: t 2: 0) is one-dimensional andY corresponds to the unit vector field

on s1), with variations [21], and gradient Brownian flows, canonical flows on

the frame bundle, and some stochastic mechanical flows, [22]. The latter are

171335

usually flows on the non-compact space IRn of the form

dXt = dBt + V log \j) dt (132)

where \j) is a smooth l2 function, with V\j) in l2, which is positive (at least for

the 'ground state' flow when it is the leading eigenfunction of Schrodinger

operator-~ t::. + V, for V a real valued function). However the estimates still

work to allow the exponents to be defined, [26], [27]; and something can be

said even in some cases when \j) is time dependent (the quasi-periodic case, for

lj) 0 a linear combination of eigenfunctions of our operator). For spaces of

constant curvature, or more generally for Riemannian symmetric spaces, the

frame bundle can be identified with p: G-+ M:::: G/H for some lie group Gas in

§5 of Chapter II, and the study of the asymptotic behaviour of the canonical

flow in this situation was begun by Malliavin and Malliavin in 1974!75 [71],

[73]. We shall look at gradient flows and canonical flows, and for the latter

look in detail at the situation for the hyperbolic plane. However we will

mainly look at it directly rather than use the gr·oup theoretic approach because

the latter is special to the case of symmetric spaces and so not likely to be

much help in obtaining results for spaces with non-constant curvature. The

group theoretic approach is carried through in detail in [10], where there is

also a very nice analysis of gradient Brownian flows on spheres.

For more recent work see [63a], [63b].

§2. Mean exponents

A. The mean exponents XL are considerably easier to study than the actual

exponents themselves. Since they are given by (128):

).. L(x) = lim 1/t log det T xF 1( -,w) t-+OO

their existence depends only on the usual additive ergodic theorem, rather than

the more sophisticated multiplicative theorem

The covariant equation (77): Dvt = VY(vt) o dzt can be interpreted after

parallel translation back to x0 and the addition of an S.D.E. for the horizontal

lift of Xt as an equation on OM x IL(IRn;IRn). To describe det TFt we can

therefore use the Ito formula, and with the notation used in §SA of Chapter II

we must calculate d/dt log det &S(t,-)at t = 0 and its second derivative. In fact

172336

it is a classical result for flows of ordinary differential equations, (the

continuity equation), which is left as an exercise, that

d/dt det oS(t,-) = div Ye(S(t,-)e).det oS(t,-).

where Ye is the vector field Y(-)e. See Lemma 2B of Chapter V.

Thus, at t = o,

d/dt log det oS(t,-) = di v Ye

and

d2/ctt2log det oS(t,-) = <V(div Ye),Ye>·

Consequently

log det T xFt = Ito L div xP(xs)dBPs +Ito div A(xs)ds

+ ~ Ito L d(V div XP(xs),XP(xs)>ds

(133)

(134)

Since M is compact L: div XP(s) is bounded, so the Ito integral in (134) is a

time changed Brownian motion B"t(t)• say with "t(t) :s; const.t for all t.

Therefore t-1B"t(t) -+ 0 as t -+ oo. Applying the ergodic theorem, with the

notation of §1A, for p almost all x:

lim 1/t log det T X Ft = J (div A(x) + ~ L <V div xP(x),XP(x))) p(dx)IP(dw) MxQ

i.e.

:>..r = JM div A(x) p(dx) +~LIM <V div XP(x),XP(x))p(dx) (135)

This is a special case of formulae by Baxendale for sums of the first k

exponents in [12 ].

For a Brownian flow, i.e. when A = ~ 6, the invariant measure pis just the

normalized Riemannian measure. We can use the divergence theorem, equation

(1.5), to dispose of the first term of (135). For the second term we can

integrate by parts: in general if f:M -+ !R is c1 and Z is a c1 vector field

applying the divergence thoerem to fZ together with the formula

div fZ = f div Z + (Vf,Z)

yields

JM f div Z dx =- JM (Vf, Z)dx

Thus

(136)

(137)

173337

A';= - (2!MI)-1 "J M '; (div XP(x))2dx (138)

where IMI denotes the volume of M.

B. From (138) we see that in the Brownian case A'; s 0 with equality if and

only if div XP = 0 for each p. More general results are obtained by Baxendale in

[12a] and we consider a simple version of those, assuming now that A is non­

degenerate.

For Borel probability measure A,Jl on a Polish space X define the relative

entropy h(A;Jl) e IR (;:: O) u + oo by h(A;Jl) = oo unless Jl s :>,and

f X dJl/dA I log dJl/dA I d),< oo (140)

in which case

h(A;Jl) := f X (dJl/dA log dJl/dA)d:>,

= f X (log dJ.l/dA) dJl (141)

To see that h(A;Jl);:: 0 observe that x-+ x log xis convex on (O;oo) so by Jensen's

inequality if h(A,Jl) < oo then

h(:>,;Jl);:: (f dJl/d), d).) log (f dJl/d), d).)= 0

with equality if and only if Jl = :>,.

We shall be particularly interested in the case where :>, = p, the invariant

measure of our S.O.S. on M, and Jl = Pt where Pt is the random measure on M

defined by

Pt(w)(A) = p(Ft(-,w)-1(A))

for A a Borel set in M and we Q.

Following Baxendale [12a] and LeJan [69] we will consider h(p;pt):

Theorem 28 For a non-degenerate system

A'; = - 1/t IEh(p;pt) (142)

Consequently:>,'; s 0 with equality if and only if p is invariant under the

sample flow t-+ Ft(-,w): t;:: 0 for almost all tJ) t Q.

Proof

Let ), denote the Riemannian measure of M, and for t ;:: 0 abuse notation

so that Pt(dx) is written Pt(X)dx: by standard results, since A is elliptic,

Pt(-,w) : M-+ IR is smooth and positive. Then p 1(x) = p 0 (y)(det T yFt)-1 for

y = Ft-l(x), and so

IEh(p;pt) =IE f (log Pt(X)- log p 0 (x))pt(X)dx

174338

= lE J (log Pt(F t(x)) - log p 0 (F tCx))) p 0 (x)dx

= lE J (log p 0 (x)- log det T xFt - log p 0 (Ft(X)))p 0 (x)dx

=- lE J (log det T xFt)p 0 (x)dx

= -tAL: by (134) and (135). I I

In [12a], Baxendale gives a formula for h(p;p 1) analogous to (134) and

using analogous computations which can be based on the continuity equation

d/dt p 0 (S(t,x)e)det oS(t,-)) = div(p 0 Y(-)e)(S(t,x)e).det o S(t,-) (143)

See equation (285) in §2B of Chapter V below.

As he points out some non-degeneracy conditions are needed: in the

completely degenerate case of an ordinary dynamical system, when p is the

point mass at a source A:L; > 0, and when it is a point mass at a sink A:L; < 0,

while in either case h(p;Pt) = 0.

c. For a gradient Brownian system (see §7 Chapter II) it is possible to get a

neat upper bound. For <p : M-+ IRm the isometric embedding, so that XP = VcpP,

equation (138) gives, by the Cauchy-Schwarz inequality,

A L: = - 1/(2!MI) J M L:p (6<pP)2dx

:S- 1/(2!MI) L: (J M cpPt:.cpP dx)2 (J M (cpP)2 dx)-1.

By a translation in IRm we can assume that J cpP dx = 0 for each p, i.e. that

cpP is orthogonal in L2 to the solutions toM= 0 (i.e. the constants, since M is

compact). Therefore, with an integration by parts

A:L; :S 1/(21MI) L: J M !XP(x)l2x dx J M cpP t:.cpP dx/ J M(<pP)2dx

:S 1/(2!MI) x (leading eigenvalue of 6) f M L:p IXP(x)l2 dx.

Since L:p IXP(x)J2 = n for all x, this yields Chappell's result [25]:

Proposition 2C Let J.l be largest non-zero eigenvalue of Li. Then

1/n A:L; :S ht. (144)

Moreover there is equality if and only if~ 6<pP = J.l<pP for each p, [25],

[22].

Such embeddings have been studied by Takahashi, see [65] Note 14. The

simplest examples are the spheres sn with their standard embeddings in 1Rn+1.

Another example is the torus in IR4 which is the image of

cp: s1 (1/2) x s1(1/2)-+ IR4

175339

~(u,v) = (1/2 cos u, 1/2 sin u, 1/2 cos v, 1/2 sin v) (145)

Here A.:L = -2.

Note that by (74), in the gradient Brownian case, (138) is just

A.:L = - 1/(21MI) I M I trace o::xl2 dx (146)

For the sphere sn(r) of radius r in 1Rn+1 this shows directly that

1/n A. :L = - ~ n/r2

since

o::x(v,v) = - ..!_ lvF (x/lx!) r

(147)

(148)

§3. Exponents for gradient Brownian flows: the difficulties of estimating

exponents in general

A. For a gradient Brownian flow, if Vt = T x F1(v 0 ), equation (95) reduces, 0

when lv0 1 = 1, to

dlvtl2 = 2(Vt,VX(Vt)dBt>- Ric(vt,Vt)dt + :Lp(VXP(vt),VXP(vt))dt (149)

giving

1/t log lv11 = 1/t It0

<1ls.VX(fts)dBs>- 1/(2t) It0

Ric(fts,fl.s)ds

- 1/t It0

:Lp <1l.s• vxP(1\s)>2 ds

+ 1/(2t) I 10

:Lp (VXP(vs),VXP(vs))ds (150)

where 1\s = Vs/lvsl in the sphere bundle SM for SxM = (v e T xM: lvlx = 1). By

(70) for v e T xM

vxP(v) = Ax(v,ep - xP(x))

so

and

Thus

176340

t

lim .!..loglvtl =lim.!.. J ~I cxx<Tls,-)12- lcxx<Tls•Tls)l2- ~ Ric (Tls•Tls)ds (151) t t

0

(almost surely).

This can be modified by the use of Gauss's theorem that for vET xM

Ric (v,v) = -1 cxx(v,-)12 + <cxx(v,v), trace cxx> (152)

so as to get an expression entirely in terms of the second fundamental form

and process Tlt: 0 ~ t <co.

so

For sn(r) in IRn+1, if u, vET xsn(r)

(n-1) 1 x Ric (u,V) = - 2- (U,V) and CXx(U,V) = - r <U,V>r

r

lim 1/t log lvtl =- ~ n/r2.

t~co

Thus

A 1 = - ~ n/r2 ( 153)

and so A1 = 1/n AL by (147). This shows that all the exponents for the spheres

are the same. Bougerol [19] has shown that among all hypersurfaces it is only

the spheres which possess th1s property.

B. The process Tlt: 0 ~ t < co 1s g1ven by an S.O.S. on SM. In fact this is

just d'Tlt = P(Tlt)&X(Tlt) o dBt where P(71) is the orthogonal projection in T xM of

T xM onto TSxM· By compactness of SM it will possess ergodic probability

measures which project onto p. If vis one of these we get for v-almost all v0

lim 1/t log lvtl = JSM nlcxx(Tl,-)12- lcxx(Tl,Tl)l2- ~Ric (Tl,Tl)v(d'Tl) (154)

As the right hand side of (154) varies over the ergodic measures v which

project onto p it gives a subset of the set lyapunov exponents, sometimes

called the Markovian, or deterministic, spectrum. They correspond to

elements of the flltration which are non-random: see [23], [62] for details. In

particular the top exponent Allies in this set.

Thus if the integrand of (154) is strictly negative for all 11 E SM the top

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exponent will be negative. Using (152) it is straightforward to see that this

holds for hypersurf aces if the pri nci pa 1 curvatUt~es 2 1 (x), ... ,.2n(x) at each point

x satisfy

x e M, j = 1, .•. ,n

for some e > 0, see [26]. This is a convexity condition. It seems reasonable

to guess that X 1 < 0 when M is the boundary of a convex domain.

B. Formula (150) is a version of Carverhill's VE~rsion of Khasminski's formula

(see his article in [B]). From it we get more general versions of (151) and

(154). A major difficulty in extracting information from (154) is the lack of

knowledge of the behaviour of the invariant measures v, in particular lack of

knowledge about their supports (one cannot expe~ct the infinitesimal generator

of 'Tlt : t 2:: 0 to be elliptic or even hypoellipt·ic in general). Control theory

gets involved here: see the article by Arnold et al in [8], and also more recent

work by L. Arnold and San Martin.

C. Rather than considering the process 11t: t 2:: 0) on SM it is often more

convenient to take its projection onto the projective bundle PM which is simply

the quotient of SM obtained by identifying antipodal points in each fibre SxM· It

is shown in [12a] that given ellipticity of A (for example) there is an invariant

measure v for this process such that with vt its shift by the flow of the

process on PM

IEh(v;v1)- h(p;p1)) ~ n X1- XL (155)

where his the relative entropy as in §28. Using this Baxendale showed that all

the exponents are equal given some non-degeneracy of A (e.g. ellipticity), if

and only if there is a Riemannian metric such that the sample flows Ft(-,w)

are conformal diffeomorphisms. See also [19].

D. For gradient Brownian flows the exponents and their multiplicities are

geometric invariants of the embedding of M into IRm. We have seen that in

general there are non trivial filtrations of tangent spaces T xM. These are

dependent on the embedding and the particular sample path: it is rather

difficult to imagine what, necessarily long time, property of the sample path

will determine the position of say v(2)(x,w) in T xM·

178342

§4. Exponents for canon1cal flows

A. Consider the canonical flow on the orthonormal frame bundle OM of the

Riemannian manifold M, (or on SOM if M is orientable, for some orientation).

There is a natural metric on OM defined by requiring that the trivialization tp

of TOM, given by equation (107) in §9 Chapter II gives isometries 'Pu: T uOM -t

IRn x .Q.(n) for each u e OM. Here the inner product on .Q.(n) is taken to be

(A,B) = - ~ trace AB (156)

for A,B e .Q.(n) identified as skew-symmetric matrices. The factor of ~ has

some advantages e.g. if e e IRn

IAel ~ IAIIel (157)

with this definition. (A disadvantage is that it was not used in [24]). The

corresponding measure on OM is sometimes called the Liouville measure.

Since T uTI : T uOM -t T n(u)M is an isometry on the horizontal subspace HuOM and

vanishes on its orthogonal complement, we see TTT maps the Liouville measure

onto the Riemannian measure of M. Also by the invariance ()oTRg = ad(g-1)offi

of connection forms and the invariance under ad(g-1) of the given inner

product on .Q.(n) it follows that the Liouville measure is invariant under the

right action of O(n) on OM.

It is a standard result, observed by Malliavin, that the canonical flow has

sample flows Ft(-,w) which preserve the Liouville measure. Rather than check

that div XP = 0 for each p we can see this from the Stratonovich equations

(121a) and (121b)

ds t = At o dBt

dAt = Ut -1 R(Ut o dBt, Ut S t)Ut

of §9C Chapter II for s t = e( oF t(V)), At = w (oF 1(V)). Indeed the equation for s t

involves only At and conversely, so the trace of the right hand side considered

as a linear transformation of <st,At) vanishes identically. Therefore the

Stratonovich equation for det oFt(-) shows that the determinant is identically

1, and so the Liouville measure is preserved.

Our Lyapunov spectrum wm be taken wHh this as basic measure. However

in general it will not be ergodic: for example it will not be if M is the product

M1 x M2 of two Riemannian manifolds, or when M is flat (i.e. has vanishing

curvature). In the latter case we noted in §98 of Chapter II that OM is foliated

179343

by horizontal submanifolds: each of these will be invariant under the flow.

More generally the holonomy bundle, see [64], is invariant.

C. Since the system can degenerate we must first show that the exponents

:>..(r)(u) < ••• < :>..( 1)(u) can be taken to be independent of u e OM. Here, and for

the rest of this discussion of canonical flows we are following [24]. To do

this observe (as in the proof of Theorem 4C, Chapter II) that forge O(n)

Ft(U,W)·g = Ft(U·g, g-1w) (158)

and so for V e TOM

TRg(TFt(V,w)) = TFt(TRg(V), g-1w) (159)

The measure one subset r of OM x Q in Theorem 18, consisting of points for

which convergence to the exponents occurs can therefore be taken to be

invariant under (u,w)-+ (u·g, g-1w) for g e O(n), with corresponding invariance

for the filtrations i.e. vm(u,w) = y(j)(u·g, g-l.w), and so for the exponents:

A.(j)(u·g) = A.(j)(u), since they are non-random. Thus we obtain maps A.(j)0 : M-+

IR with A.(j) 0 (n(u)) = A.(j)(u) for u in OM, defined almost surely. These are

measurable. Also since each A.(j) is invariant under ~t:OM x Q-+ OM x Q, we

have

:>..(j)(u) =IE :>..(j)(Ft(u,w)) =IE :>..(j) 0 (n F1(u,w))

=Pt:A.(j)o(X)

for x = n(u), where (Pt: t 2: 0) is the heat semigroup forM (solving o/ot = it.).

Thus Pt A.(j)0 is independent oft, and so A.(j)0 is constant (for example by the

ergodicity of the Riemannian measure: but this itself is usually proved by

observing that Ptf independent of t implies 6P1f = 0 for t > 0, since Ptf is c2

for t > 0, which implies Ptf is constant for each positive t , which implies by

strong continuity of Pt in t that f is constant).

D. From our equations (122a,b) forst= e(&Ft(V 0 )) e IRn and At= ro(&Ft(V0 )) e

o(n.) we could write down an expression for logl&Ft(V 0 )1 = i log(lst12 + 1Atl2).

However that does not seem very illuminating, and we shall resist doing so

(but see equation (172) below when dim M = 2). To start with we shall just

consider the horizontal component St· For this set Vt = Tn(&Ft(V 0 )), so Vt =

ut<st) and Vt e T xtM for Xt = n(u 1) the Brownian motion induced on M. In

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particular lvtlxt = l~tl· By (122a)

1~ 1 12 = 1~0 12 + 2 It 0 <~s• AsdBs>- I\ Ric(vs,Vs)ds + 2 I\ lAsF ds (160)

log l~tl =log 1~ 0 1 + It0 <~s/l~sl, As/l~sl dBs>- ~ It0

Ric(vs/lvsl, Vs/lvsl)ds

+It 1Asi2/J~sl2 ds- It 1As~sl2/l~sl4 ds (161) 0 0

(at least until the first hitting time "t of 0 by ~t).

Therefore by (157) and the observation that !Ael =!AI lei when n = 2

log l~tl2: log 1~ 0 1 + Mt- ~ It Ric (vs/lvsl, Vs/lvsl)ds (162) 0

with equality when n = 2, where Mt: t 2: 0 is the local martingale

t Mt = I

0 <~ sll~ sl, As/1~ sl dBs> ( 163)

Now Mt: T 2: 0 is a time changed Brownian motion and for ~t to vanish in

finite time "t (assuming ~ 0 * 0), we would have to have

limt-n- Mt = -oo. Then limt-+"t- Mt = oo and so

limt-+"t- log(loFt(V0 )1) 2: limt-+"t- log l~tl = oo

which cannot be true for finite "t. Thus l~tl never vanishes and (161) holds for

all time.

Theorem 40 [24]. Let Ric(x) = sup Ric(v,v) : v e T xM and lvl = 1 for each

x e M. Then the top exponent A 1 of the canonical flow satisfies

A 1 2: lim t-+oo 1/t log lvtl2:- 1/(21MI) I M Ric(x)dx (164)

Proof: Since Mt is a time changed 1-dimensional Brownian motion

limt-+oo 1/t Mt ~ 0 ~ limt-+oo 1/t Mt

Therefore by (162)

limt-+oo 1/t log lvtl2: lim(- 1/(2t) J1 Ric(vs/lvsl, Vs/lvsD) ds)

0

2: - 1i m 1/(2t) f t Ric(xs)ds 0

= - 1/(21MI) I M Rlc(x)dx

almost surely, by the ergodic theorem, since (1/IMI x the Riemannian measure)

is ergodic for Brownian motion. II

181345

Remark 40 For dim M = 2 the Ricci curvature is essentially the Gaussian

curvature Kp(x) for each x. The Gauss-Bonnet theorem states that

1/(211) I M Kp(X)dx = x(M) (165)

where X(M) is the Euler characteristic of M, a topological invariant (e.g. X(S2)

= 2, xcs1 X s1) = 0). It is proved in Chapter VI below. From this, (162), and

the argument above: if dim M = 2 then for p 0 G' almost all (x,w)

A 1 2:: iiillt-+oo 1/t log lvtl = limt-+oo 1/t Mt - ( 11/!MI) X(M)

;::: - (11/IMI) X(M) (166)

Also

limt-+oo 1/t log lvtl =lim 1/t Mt - (11/IMI) X(M) :S -(11/IMI) X(M) (167)

(The~ in the corresponding formula in [24 J should not be there).

From (167) we get

A1 =lim 1/(2t) log(l~tj2) + 1Atl2 =lim 1/t (logl~tl + ~ log(1 + 1Atl2 /l~tl2 ))

:S -(11/IMI)X(M) + iilli 1/(2t) log(1 + !Atl2/lstl2 ).

Since A 1;::: 0 this shows: for dim M = 2

lim 1/t log (1 + 1Atl2/lstl2);::: (211/IMI) X(M). (168)

E. Next we consider the case dim M = 2 in more detail.

Write k(x) for the Gauss curvature Kp(x) (with P = T xM) so that

Ric ( u,u) = lul2 k(x) (169a)

and

R(u,v)w = k(x)((w,v)u- (w,u)v (169b)

for u,v,w in T xM· The following formulae are given for completeness. They

come from (160) and (122b): the rather straightforward proof is left an

exercise; there are details in [24] (using the scalar curvature S(x) = 2k(x))

Is tl2 = lsol2 + 2 It 0 <s s•AsdBs> - f t

0 k(xs) Is sl2 ds + 2 f \ 1Asl2 ds ( 170a)

IAtF = IA 0 12- 2 J1 k(xs)<ss.AsdBs>- 2ft k(xs) 1Asl2 ds

0 0

+ J1 dk(usAsss)ds + J1

k(xs)21ssl2ds (170b) 0 0

The following formulae from [24] are useful:

Proposft1on 4E

For dim M = 2 there is the Stratonovich equation

182346

diAtl2 + k(Xt) o dis tl2 = 0 ( 171)

and the Ito equation

IA1l2 + k(x1)1s 112 = IA 0 12 + k(x 0 )1s 0 12 + f t Is sl2 dk(usdBs)

0

+ ~ J1 lssl2 ~k(xs)ds- J1

dk(usAsss)ds (172) 0 0

Proof The Stratonovich equations (121a,b):

give

dst =At 0 dBt and dAt = Ut-1 R(ut o dBt, Utst)Ut

dis 112 = 2<s t· At o dB1>

d1Atl2 =-trace At Ut- 1 R(Ut o dB, Uts1)ut

!. (Ut -lR(Ut o dBt.Uts t)Utep,A1ep) p=l

(173)

!. k(Xt) <uteput,st><utodBt,utAtep> - <ut ep, ut o dBt><ut s t•Ut At ep> p=l

(174)

Equation (171) follows immediately. On integrating it by parts and then using

the Ito formula for k(xt) (and hoping the use of d for stochastic differentials

as in d(k(xt)) and for ordinary differentials as in dk will not cause confusion):

IA112 + k(xt)lstF- IA 0 12 + k(x 0 )ls 0 F

= Jt d1Asl2 + k(xs) o dlssF + lssl2 o d(k(xs))

0

= ft lssl2 0 d(k(xs)) 0

= J t Is sl2 dk(us o dBs) + ~ "f t Is sl2 ~k(xs)ds + ~ f t dis sl2dk(usdBs) 0 0 0

giving (172) by (173). II

Theorem 4E

When dim M = 2 and k(x) > 0 for all x

)...1 s 1/(41MI) fMlVk(x)l/.fk(x) + l~k(x)l/k(x)dx (175)

f.r..Q.Qf

Since k(x) > 0 we can take .fIAI2 + k(x) Is 12 as the norm of (s,A) when

computing the exponents. Write it as ll(s,A)I!. By (172)

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log ll<st.At)ll =log ll<s0 ,A0 )11 + ~ Jt lssl21 ll(ss.As)ll2 dk(usdBs) 0

+ 114 f t 0

Is sl2 I ll(s s•As)ll2 t.k(xs)ds - ~ f \ dk(usAss s)dsiiKs s•As)ll2ds

- 114 J t Is sl4 l(dk)x 12 lll(s s•As)ll4ds ( 176) 0 s

Since

ldk(usAss s)l = (Vk(xs),usAss s> :s 1Vk(xs)IIA5 lls sl

:s ~ 1Vk(xs)l.fk(xs)-111(s s•As)ll2 (177)

and the coefficient of the Ito integral in (176) is bounded, the ergodic theorem

gives the result. I I

This corrects the upper bounds in [24 ]. It must be possible to do better.

F. Next we consider the case of constant curvature. An important point here,

and later, is the idea of a covering p: M-+ M. This is a C00 map of manifolds

which is surjective and such that each x e M has a connected open

neighbourhood U with p mapping each component of p-1cu) diffeomorphically

onto u. The typical examples are p: s1-+ s1 given by p(eie) = e2i9 and

p:IR -+ s1 given by p(e) = ei e. The covering is Riemannian if M and M are

Riemannian and T zP: T 2 M-+ T p(z)M preserves the inner product. Clearly if M

is Riemannian and p is a covering map then we can define a Riemannian metric

on M so that p becomes Riemannian. In general coverings have the path

lifting property; if d : [a,b)-+ M is a continuous path and z e p-1(d(a)) then

there is a unique continuous d- : [a,b) -+ M with d-(a) • z and p o d- = d. The

lifting gives a continuous map from the space or continuous paths in M starting

from <5(a) to the corresponding space of paths from z in M. Thus stochastic

processes can be lifted from M toM. Also it is easy to construct an S.O.S. -(Y ,z) p-related to a given one (Y,z) on M.

For a Riemannian covering a Brownian motion on M maps by p to a

Brownian motion on M: to see this choose an S.O.S. (Y,z) on M which has -Brownian motions as its solutions. The lift (Y ,z) will then have Brownian

motions on M as its solutions since the condtions of §4A Chapter II for this to -happen are purely local, and locally (Y ,z) and (Y ,z) are the same, as are M and -M. Since (Y ,z) and (Y,z) are p-related the result follows by §3D Chapter 1.

(For a generalization of this see [43], p. 256.) Since curvature is a local

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property p will also map the curvature tensors of M to that of M.

Given the equation dxt = X(xt) o dBt + A(xt)dt on a compact Riemannian M

and a Riemannian covering p : M -+ M, it is immediate that the Lyapunov

filtrations and any stable manifolds lift to corresponding objects for the lift of

the SDS: the filtration ofT 2 M will map by T zP to the filtration forT p(z)M and

will exist when the latter exists, with the same exponents, and p will map

stable manifolds to stable manifolds as a covering. In particular Nl. need not be

compact.

The other two main ingredients we need are:

(i) there exists a covering p : Nl. -+ M with Nl. simply connected, and this is

essentially unique

(ii) if M is simply connected and of constant curvature k then: M is isometric

to IRn if k = 0, to sn if k = 1, and to hyperbolic space Hn if k = 1. (e.g. see [64 ]).

If we now note that for a Riemannian covering p: M-+ M the map u-+ Tpou

:OM-+ OM is also a covering, we see that to investigate the Lyapunov exponents

and stable manifolds when M has constant curvature k = +1 or k = -1 it suffices

to take M = sn or M = Hn.

G. Suppose now that M has constant curvature k. Equations (173) (174) are

valid with k(xp) replaced by k, because of formula (9) for the curvature. Thus,

[24 ],

d!Atl2 + k dl~tl2 = 0

whence

(178)

When k > 0 we see immediately that :>-.1 = 0, whence :>-.1 = A.L:, and so all

the exponents vanish given constant positive curvature.

Fork< 0 we see that except perhaps for some exceptional V 0

:>-. 1 =lim 1/t log lstl =lim 1/t log IAtl

t-+oo t-+oo

and in particular these limits exist.

From ( 164) this yields

lim 1/t log lstl ~- (n-1)k/2 > o t-+OO

185349

so that almost surely

~ 1Atl211stl2 = -1< + ~ 0A0 1 + 1<ls0 12)11stl2 = -1<

From (161) (162) we see 1mmed1ately that if .Q.Lm M = 2 then for k < o (179)

G. For constant negat1ve curvature since ).1 > 0 and A:E = 0 there must be

some negative exponent and correspondingly sCtme stable manifolds. We will

investigate these for k = -1 and dim M = 2 following [24]. By the discussion in

the previous paragraph we need only take M to be the hyperbolic plane H2, even

though it is not compact.

In §SC of Chapter II we described H2 as the hyperboloid (t,x,y) e IR3: t ~ 1

and x2 + y2- t2 = -1 with Riemannian metric induced from the Lorentz metric

of IR3.

Writing N = (1,0,0) the tangent space TNH2 can be identified with

0) x IR2 in IR3. For v = (v1, v2) in IR2 the path 'tv: IR-+ H2 given by

"tv(<X) = (cosh(lvi<X), v1/lvl sinh(lvi<X), v2/lvl sinh(lvi<X))

has

d~ lv(<X) = (lvl sinh(lvi<X), v1cosh(lvi<X), v2 cosh(lv!<X))

so that

d 1-d "tv(<X)I ( ) = lvl.

a lv <X

Differentiating this we see D/o<X d/d<X "tv(<X) is orthogonal to d/d<X "tv(<X) and

so vanishes by symmetry. Thus "tv is a geodesic through N and using our

identification of TNH2 wHh IR2

expNv = l v< 1).

For V e T viR2 ::::: IR2 • with v * 0

DexpN(v)(V) = (<v,V)/Ivl sinhlvl, (sinhlvl)DP(v)(V) + (<v,V)/!vl)(coshlvl)(v/lvl))

E IR X IR2

where P: IR2- 0)-+ s1 is P(v) = v/lvl. Thus

! D expN(v)(V)J2 = (V ,P(v))2 + IDP(v)(V)j2(sinhlvl)2

186350

since DP(v)V is orthogonal to v in IR2. This gives the induced metric in the

chart given by expN i.e. in normal co-ordinates at N

(V,W>v = (V,P(v))(W,P(v)) + sinh(lvi)2(DP(v)V,DP(v)W)IR 2 (180)

which is most easily considered in polar co-ordinates: in classical notation

ds2 = dr2 + (sinh r)2de2 • (181)

(One way to see this is to interpret (18)) as meaning that if a curve 6 in

M is given in normal co-ordinates by 6(t) = Crt cos et , rt sin et) then

lo(t)G(t) = r; +(sinh rt)2 ·a;).) Let oo be the open unit disc in IR2. Define f: D-+ IR2

by

f(r,e) = (2 tanh-1r,e)

in polar co-ordinates. The metric (181) on IR2 induces the metric

ds2 = 4(1-r2)-2 (dr2 + r2de2) (182)

on oo, or in Cartesian co-ordinates

ds2 = 4(1-r2)-2 (dx2 + dy2). (183)

The disc with this metric is the Poincare disc model of H2. It represents N as

(0,0), but since the subgroup G of the Lorentz group acted transitively on the

hyperboloid as isometries we can compose expN of: D0 -+ H2 with an isometry

to get an isometry which maps (0,0) to any given point of H2.

There is also the representation of H2 by the upper ~-plane

u = (x,y): y > 0 in IR2. For this choose some point c of s1. Then there is an

analytic diffeomorphism de: U-+ oo

dc(z) = c(z-i)/(Z+i) (184)

which maps the closure u to the closed disc D with the point at infinity in U

mapped to c. The metric induced on U is

ds2 = y-2(dx2 + dy2) (185)

The disc model shows how to talk about "points at infinity" on H2 :they can

be taken to be the points of s1. For c e s1 write uc for U when de has been

used to give it its metric.

H. Since any p e H2 can be identified with (0,0) in D we can identify the points

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of 'the circle at oo' , N00 , in H2 with ~ rays 1 emanating from p and

parametrized by arc length. The Buseman function of such 1 corresponding to

cis

J3p (c,-): H2-+ IR

for

J3p(C,Z) =lim (t - d(Z,"t(t))) 1-+00

(186)

(Since t -+ t - d(z,"t(t)) is increasing and bounded above by d(z,p) this limit

exists). This is sometimes given the opposite sign.

In the model uc with p = (0,1) we have "t(t) = (o,et) and if z =

(x,y) e uc then

t- logy :S d(z,"t(t)) :S t-log y + e-2t lxl

since logy is the distance of z from the line (cx,1): ex e IR. Thus in this case

J3p(c,z) =logy (187)

Lemma 4H [24]

Let zt : t 2:: 0 be a Brownian motion starting from p. Then with

probability 1

and

(i) z00 (w) =lim Zt(w) e N00 exists t-+OO

(ii) lim 1/t J3p (z 00 (w), Zt(w)) = ~ t-+OO

(188)

Proof Part (i) is a very special case of Prat's result for not necessarily

constant curvature. In our case it follows because in oo our Brownian motion

is just the time change of an ordinary Brownian motion in IR2, and the latter

almost surely leaves 0° in finite time.

For (ii) it is enough to show that

IP1/t J3p(C,Zt(w))-+ ~ I z00 (w) =c)= 1.

To condition z to tend to c we can use the Doob h-transform. Now, as

described in [81] 2X9 the standard Brownian motion IR2 conditioned to exit

from oo at a point c of s1 is the h-transform of that Brownian motion, h­

transformed by the Poisson kernel

h(z) = (1- lzF)/Ic-zl2 lzl < 1.

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This means it has the law of the diffusion process with generator ~t:. + 'Vlog h

for t:., 'V the Euclidean operators. Since time changing commutes with our

conditioning the hyperbolic Brownian motion of M conditioned to tend to c as

t-. oo is a diffusion process with generator ~t:. + 'V log h where now t:. and 'V

refer to the hyperbolic metric.

In the model Uc the Laplacian is given, for x1 = x, x2 = y, by

M(z) = 'f g(z)-112 a;axi g(z)~ gii(z) of /oxi J=l

= y2 (o2f ;ax2 + a2f ;ay2)

while his represented by h ..... for h ..... = h ..... • de i.e.

h ..... (z) = 1/4(1z+il)2- lz-il2) = y for z = x + iy.

Thus

V log h ..... (z) = (0, y2. 1/y) = (O,y)

(189)

and the conditioned diffusion can be represented by Zt = (Xt•Yt) for

dx t = Yt dB 1t, d!Jt = Yt dB2t + Yt dt (190)

where (B 1t, s2t): t 2:: 0 is a Brownian motion on !R2. Then Yt =Yo exp(B2t +

~ t) and so (ii) follows by (187). I I

We can now give the basic result from [24] on the stable manifolds of the

canonical flow on OM for hyperbolic space:

Theorem 4H [24] For M = H2 take u e OM. Let Ft(-,w):OM -+ OM be the

canonical flow. Then for almost all w e Q the following holds:

The limit c(w) =lim TT F1(u,w) exists in N00 and if 'Y'(u,w) is the 1-+00

submanifold of OM given by Tg o u s.t. g: uc -. uc is a horizontal

translation) then for u' e V(u,w)

lim 1/t log d(Ft(u,w), Ft(u',w)) = -~ t-+OO

and for all other frames u'

lim. 1/t log d(Ft(u,w), Ft(u',w)) 2:: 0

t-.oo

(191)

(192)

Proof: Choose w e Q so that the conclusions of Lemma 4H are true, and so that

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the flow Ft(-,w) exists and satisfies Ft((Tg) o u,w) = Tg o Ft(u,w) for all

isometries g of M. The latter is possible either by general principles, because

the canonical S.O.S. is invariant under the action of such Tg, or by noting the

special properties of the flow Ft(-,w) on OM when OM, or rather SO(M), is

identified with our subgroup G of the Lorentz group: see Remark 4H(i) below.

Then c = z00 (w) exists. We will work in uc.

It is necessary only to consider oriented frames i.e. restrict ourselves to

the component SOM of OM. Such a frame at (x,y) e uc can be identified with a

tangent vector to uc of unit Euclidean length. Using this we shall write frames

as (x,y,A) e uc x s1. Let d"' be the metric on SOM which is the product of the

Euclidean metric on uc with the standard one for s1. Over the compact subset

W of uc

W = (x,y) e uc: lxl + 11-yl ::s ~ this will be equivalent to the standard metric of OM described previously (or to

any other metric).

Set (Xt,Yt,At) = Ft(u,w).

If u' e 'V'(u,w) there exists a e IR with

F1(u',w) = (Xt +a, Yt•At) t ~ 0

Horizontal translation in uc is an isometry and so is the dilation (x,y)-+ (exx,

exy) for ex> 0. Therefore

d(Ft(u,w), Ft(u',w)) = d((O,yt,At), (a,yt,At))

= d((0,1,At), (a/yt, 1, At)) (193)

since isometries on M induce isometries on OM. For sufficiently large t both

(0,1) and (ayt-1,1) lie in W, and sod may be replaced by d in estimating (193)

for such t. However .....

d ((0,1,A1), (a/yt, 1, At))= lal/lytl

and by Lemma 4H(ii) and equation (187)

1 i m 1/t log (lal/lytl) = - ~ t-+oo

This proves ( 191).

(194)

For (192) first suppose u' = (exx0 , exy0 , A0 ) for some ex> o, ex* 1 where u =

190354

(x 0 ,y0 ,:>.. 0 ). Then F t(u' ,w) = ( «XXt, o:yt,:At), giving (by a hori zonta 1 translation)

d(F1(u,w), F1(u',w)) ~ d((O,o:yt), (O,y1))

= d((O,o:), (0,1))

from which (192) follows.

Combining this with (191) we see the same holds for any u' which is

obtained from u by the action of the isometries of M generated by the

horizontal translations and the dilations (x,y) ... (o:x, o:y) of He. These

isometries correspond to a subgroup Gc, say, of G when we use the

identification of SOM with our subgroup G of the Lorentz group. The group G

itself was identified with isometries of Min the hyperboloid model and in this

model it is easy to see that Gc is precisely the subgroup of G which leaves the

point at infinity c fixed (the latter subgroup is just the natural embedding in G

of the identity component of the Lorentz group of the 1 + 1-dimensional space

time acting in the plane orthogonal to c in IR3: this is two dimensional as is Gc

and the former is known to be connected).

For other u' in SOM = G there is the isometry corresponding to

g = u·u-1 which sends u to u'. Since Ft(u',w) = g Ft(u,w) and g is not in Gc

lim Ft(u',w) *c. 1-.00

Consequently

d(Ft(u,w), Ft(u',w)) ~ d(TT Ft(U,W), TTFt(u',w))

~ d(TT Ft(u,w), (x,1):x E IR))

or sufficiently large t. This is just J3p( c, TT ,F 1(u,w )) for p = (0, 1) by ( 187). Thus

in this case, by ( 188)

lim 1/t log d(Ft(u,w), Ft(u',w)) ~ ~. II

This theorem, together with the fact that we know there must be at least

one negative exponent with corresponding stable manifolds, shows that there

is precisely one, namely-Land that the stable manifold through u is Y(u,w).

Consequently the multiplicity of the exponent - ~ is dim'V'(u,w) i.e. 1. Since dim

OM= 3, and :>..:L = o and :>..1 =~by (179), because 2(~) + (- ~) = ~ * o there must

be another exponent. It can only have multiplicity 1 and it must be 0. Thus

the exponents for the canonical flow on H2 are-~. 0, ~·

As for the filtration of TuM we know that v(3)(u,w) = T uY(u,w). It will

191355

now be no surprise that y(2)(u,w) is the tangent to the orbit of Gc (i.e. the

tangent to the coset Gcu in G): for a proof see [24]. More detailed information

'stability' properties of the flow can be found in [24] and [10], especially the

latter.

Remark 4H(j) Identifying SOM with G the canonical S.D.E. becomes a left

invariant stochastic differential equation

dUt = X( Ut) o dBt

with X(1)(0 = .t for s in IR2 as in equation (56) of Chapter II §SB. The flow is

then Ft(u.w) = u.g1(w) where gt : t 2: 0 is the solution starting from 1.

Equation (178) showing that IAtF- lst12 is constant follows from the invariance

of the Cartan-Killing form: see [64] p. 155. The metric we have taken on SOM

corresponds to a left invariant metric on G so our exponents are measuring

how right multiplication by gt(w) spreads out or contracts the space (at least

infinitesimally). Use of the Lie group structure of G gives a good way to obtain

the result about the exponents given above and especially for their higher

dimensional analogues. This is carried out in [10 ]. See [71 ] , [73] for

earlier work for symmetric spaces.

The vanishing of the exponents for sn comes out particularly simply by the

corresponding representation of sosn as SO(n+1). This time the metric on

SO(n+1) is bi-invariant and so F1(-,w) consists of isometries: this is the reason

for the constancy of lst12 + 1Atl2 in equation (178) fork= 1.

Remark 4HOO The projections onto M of the stable manifolds 'V'(u,w) are

horocycles. In the disc model the horocycle Hp(c) for p e M and c on the circle

at infinity is the circle tangent to s1 at c which goes through p. The

horocycles are precisely the level surfaces of the Buseman functions defined

by (186). Equivalently they can be defined as the boundary of the horoballs

defined as the union U Bt( l(t)) of balls radius t about '¥( t) for l a unit t>O

speed geodesic. These definitions make sense in greater generality: in

particular for simply connected manifolds of non-positive curvature. For

more details see [2], [8]. However there is no reason to believe that stable

manifolds for the canonical flows of these more general manifolds project

onto these horocycles.

Remark 4H(iii) For results about the non-triviality of the spectrum for the

192356

canonical flow when M * sn see [24a].

Remark 4HC1y) The characteristic exponents for the geodesic flow on the unit

sphere bundle in TM have been studied a lot [2 ], [106]. The results are

analogous for constant negative curvature: especially for dim M = 2 when the

bundle SOM can be identified with the sphere bundle. See also [105].

§5. Moment exponents

In this section we no longer require M ito be compact.

A. Consider a process Cxt : t ~ 0 on M and a process vt : t ~ 0 on some

space 8 with projection p : 8 -+ M such that p(vt(w)) = x1(w) for t ~ 0. If the

fibres p-1(x) of 8 are normed vector spaces for each x e M we can consider

for q e IR. Typical cases of interest are:

(i) 8 = M x IR and p the projection with Vt defined by

dVt/dt = V(Xt)Vt

(195)

(196)

for given v0 , for V: M-+ IR. We should then write Vp as vp(v0 ) etc. This is the

situation of the "Kac-functionals" studied extensively in [15], [54], [55]

especially in the non-compact case, i.e. the behaviour as t-+ oo of

I t V(x )ds 0 s

1/t log IE e (for v0 * O).

(ii) The analogue of (i) for 8 = M x IRn and V: M-+ n..(IRn; IRn)

(iii) p: 8-+ M the tangent bundle or a tensor bundle like APT*M with Vt defined

by a covariant equation

Dvt/ot = V(x1)vt (197)

where V(x) e n..(p-1(x), p-1(x)) for each x in M. One could equally well take

other vector bundles over M with a linear connection: this would then include

(ii) as the special case of the trivial bundle.

(iv) Vt = T x0

Ft(v0 ) where Ft(-,w) : t ~ 0, w e Q) is the flow of an S.D.E. on

M.

The last example is somewhat more complicated than the previous ones

since the equation for Vt is a stochastic differential equation in general. We

shall look in more detail at situations related to cases (i) and (iii) in the next

193357

chapter. Case (il) was investigated in [3]: there is the following general

result essentially taken from there as in [8]. In case (iv) it relates these

moment exponents to the Lyapunov exponents of the flow.

Proposttfon SA

Let v- and v _ be the random variables

v- = Tiiii1

11t log lvtl -+oo

and v- = lim.t-+oo 1/l log lvtl

Then

(i) q-+ vq is convex

(ii) q-+ 11q Vq is increasing

Also if IEiv-1 < oo and IEiv _I< oo

(iii) Vq;::: lim 11t IE lvtlq;::: q lEv_ q;::: 0

Vq2:QIEV_ q;:::O

d d -vql

0 ::s; IE v_ ::s; -vql

0 .

dq- q= dq+ q= (iv)

f.r:.2.Q1 [3] Part (1) comes from the convexity of q-+ log IE IZIQ for any random

variable Z and (ii) comes from the monotonicity of q-+ (IEIZIQ)ilq for q > 0 and

of q-+ (IE(11IZ!)-Q)- 11q for q < 0. Also by Jensen's inequality

11t log IE lvtlq ;::: IE 11t log lvtlq .. q IE 11t log lvtl

if log lvtl is integrable, so that (iii) and hence (iv) follows by Fatou's

lemma. II

8. To show one reason for studying the moment exponents let us go back to

the canonical flow on OM of a Riemannian manifold. Assume it is

stochastically complete so the solutions of the canonical S.D.E. ex1st for all

time. There is then the formal derivative flow which can be represented by

(s1, At) e IRn x Q.(n), as before, satisfying (121a,b) and (122a,b). Using the

notation of §4 set vt = UtSt· We can consider a 1-form cp on M as a section of

T*M or as cp: TM-+ IR with the restrictions fPx: T xM-+ IR linear. The element in • T xM dual to fPx wm be written fPx •

194358

Lemma 58

For a c2 1-form q> on M

t t q>(Vt) = q>(v0 ) + f Vq>(Usd8s)Vs + f q>(UsAsd8s)

0 0

1 t 2 . * + 2 J (trace V q>(Vs)ds - R1c(q>x, Vs))ds

0 s

(198)

Proof Coming back to the S(t,u)e and <SS(t,V)e notation of §98 Chapter II recall

that by Lemma 98 of Chapter II if J(t,V) = Tn(&S(t,V)) then J(-,V) is a Jacobi

field with Dlot J(t,V) = uro(V)e at t = o. Therefore, with 1(t) = nS(t,u)e,

dldt q>(TnoS(t,V)) = Vq>('Y(t))(J(t,V)) + q>(Diot J(t,V))

= Vq>(ue)(TnV) + q>(uro(V)e)

at t = 0; and, at t = 0,

d21dt2 q>(Tn&S(t,V)) = v2q>(ue,ue)(TnV) + 2Vq>(ue)(uro(V)e)

- q>(R(TnV,ue)ue).

Now for an orthonormal basis e1, ... ,en of IRn if f; = uei and S is the skew

adjoint operator uro(V)u-1

• Li Vq>(ue1)(uro(V)e;) = Li Vq>(f1)(Sf1) = Li (Vq> (fi),Sfi)

=-trace sv•q> =- trace(V.q>)S • =- Li ((Vq> )S fi, fi) =- L Vq>(Sf;)fi

=- L dq>(Sfi,fi)- L Vq>(fi)Sfi.

Thus (198) holds by Ito's formula. I I

The (de Rham-Hodge) Laplacian l:::.q> of a 1-form q> satisfies the

Weitzenbock formula

l:::.q> =trace v2q>- Ric(-,q>•) (199)

(with non-standard sign conventions), see Proposition 3D of Chapter V, below,

for the proof. The following result is discussed in [75], [43], [77].

Theorem 58 Suppose the family q>t: t ~ 0) of 1-forms on M satisfies:

(i) q>t is c2 on M and c1 in t, with the partial derivatives jointly

continuous,

(ii) oq>tlot = ~ t::.q>t t > o

195359

(iii) d<J>t = 0 t > 0

(iv) 'Pt is bounded uniformly in t e [O,T] each T > 0.

Assume M is stochastically complete and lvtl lies in L1 for each t where

Vt = ut st for so= uo-1v0 any frame u0 at x0 , and A0 = 0.

Then

!9t(v0 ) = IE<p 0 (vt) V0 E T x M, t ~ 0. 0

Proof Set 'Pt = IPT -t for 0 :S t :S T and apply the time dependent version of

Lemma 58 to 'Pt· I I

Note that (iv) holds automatically if M is compact as does the integrability

of lvtl. It is also true that d<p 0 = 0 implies d<pt = 0 when (ii) is satisfied, at

least for M compact. Furthermore, as we will see below, t::,<p = o implies d<p =

o forM compact. Thus

Corollary 58(1). If M is compact and li mt IE !vtl = 0 there are no -+00

harmonic 1- forms except 0. I I

Note that from the analogue of (170a) for constant curvature k if we

substitute klssl2 + 1Asl2 =owe see

1Eivtl2 = 1Eistl2 = e-3kt Ivai

and so the conditions of 5B(i) hold if k > 0. By (160) we have

t t iE lvtl2 = ls0 12 - f IE Ric (vs,Vs)ds + f IE 1Asl2 ds

0 0

so its hypotheses cannot hold if Ric (v,v) < o:lvl2, for all v, for some ex< 0. On

the other hand if the Ricci curvature is strictly positive everywhere Bochner's

theorem implies that there are no non-zero harmonic 1-forms. This will be

discussed in detail below.

C. For more about moment exponents and also their relationships with large

de vi ati on theory see [3 ], [ 4 ], [ 11 ], [ 13], [25], [ 45].

196360

CHAPTER IV. THE HEAT FLOW FOR DIFFERENTIAL FORMS AND

THE TOPOLOGY OF M.

§1. A Class of semigroups and their solutions.

A. Let p : 8 _. M be some tensor bundle over a Riemannian manifold M e.g. 8 =

TM, T*M, APTM, or a trivial bundle M x IRn, with induced inner product on each

Bx := p-1(x) (in fact any Riemannian vector bundle with a Riemannian

connection would do). For x e M suppose we have a linear map Jx: 8x _. Bx

depending measurably on x. Let x 1 : t ~ 0) be Brownian motion on M from the

point x0 : we will assume M is stochastically complete.

For v0 e Bx define the process vt; t ~ 0 over xt: t ~ 0 by 0

Dv1/ot = Jx1<v1) (200)

as in equation (47). Assuming J is bounded above (i.e. the map j defined below

is bounded above) the solution of (200) will exist for all time and

d/dt lv1!2 = 2 <Jx1<vt),vt>xt (201)

::; 2 j(xt) lvtl2 (202)

if j(x) = sup <Jxv,v) : v E Bx and lvl = 1.

Thus

l

J j(xs)ds

lvtl::; e 0 (203)

By a c2,1 section fPt : t ~ 0 of B* we mean a time dependent section of

the dual bundle to B: so fPt x E ll...(Bx;IR) for x e M and the map <p : 8 x [O,oo) 4 IR , . given by (v,t) 4 !flt(Y) has two partial derivatives in the first variable and one

in t, all of them continuous. The following can be considered as a uniqueness

result:

Proposition 1A. Suppose fPt : t ~ 0 is a c2,1 section of 8* such that

ocpt/ot = ~ Trace v2<pt + J*(cpt) (204)

with fPt bounded (i.e. CI~Pt xI: x eM bounded) uniformly on each 0::; t::; T, , forT > 0 • Then if J is bounded above, and v0 e Bx some x0 e M

0

197361

C9t(Vo) =IE C9o(Vt) (205)

where (vt: t ~ 0 is the solution to (200).

Proof: To interpret (200) we can suppose Xt = n(ut) for (ut : t > 0) a solution

to the canonical S.D.E. on OM. For S(t,u)e as before (e.g. §SA Chapter II) and

6 t = nS(t,u)e suppose

Dlot vt(e) = J.n<vt(e))

with v0 (e) = w 0 some w 0 e s60

• Then for tp : 8-+ IR of class c2 and linear on

the fibres, if x = n(u) = lo

dldt tp(vt(e)) = Vtp('Yt)Cvt(e)) + tp(J.'tt(vt(e)))

and, at t = 0,

d21dt2 tp(Vt(e))

(206)

= V2tp(ue,ue)(w 0 ) + 2Vtp(ue)(Jx(w 0 )) + tp(V J(ue)(w0 )) + Jx(Jx(W0 )) (207)

if J is differentiable. At first sight it is not obvious how to interpret this to

obtain the Ito formula for tp(vt) using Proposition 3A of Chapter I. In fact our

system does fit into that result but with Zt = (Bt,t), and as a system on OM x

Bx , namely 0

dUt= X( Ut) o dBt

dwt = llt- 1 J11 (ut) UltWt)dt

where I It is parallel translation of the tensors along (n(us): 0 :$ s :$ t: (this

is p(u 0 ut-1) for a suitable representation p of O(n) on Bx ). Thus the terms 0

above without adequate e's will have a 'dt' in Ito's formula and the terms

involving VJ will have a 'dt dsi 1• and so not appear. The assumption of

differentiability of J is therefore not needed (remember the global Ito formula

depends on local formulae, the way we are working out its coefficients is just

formalism: a method of obtaining a formula whose coefficients have geometric

content). Thus

t t

tp(Vt) = tp(Vo) +I Vtp(UsdBs)(Vs) +I (~ trace v2tp(Vs) + tp(Jx (Vs)))ds. (208) 0 0 s

Alternatively this can be derived from the Stratonovich equation

198362

t t

ttJ(Vt) = ttJ(v0 ) + J VttJ(Us o dBs)(vs) + J ttJ(Jx (vs))ds 0 0 s

(209)

The result follows by applying the time dependent form of (208) with 'tit =

<l'T -t where T > 0. I 1

Versions of (208) for more general systems than (200) are given in [14].

B. After a 'Feynman-Kac type' formula here is a 'Girsanov-Cameron-Martin'

formula. Let A and Z be c1 vector fields on M. Let (xt: t 2:: 0 denote Brownian

motion on M from x0 with drift A , assumed non-explosive, and (ut : t 2:: 0 its

horizontal lift to OM: so we can take it that -dUt = X(u1) o dBt +A (ut)dt (210)

-for (X,B) the canonical S.O.S. on OM and A the horizontal lift of A, with n(ut) =

Xt· Let Mt be the process on IR given by M0 = 1 and

dMt = Mt<Z(xt)•ut o dBt)- ~ Mt div Z(Xt) + IZ(Xt)l2dt

so

t t

Mt = expcJ (Z(xs),usodBs> - ~ J (div Z(Xs) + IZ(xs)l2 ds 0 0

In the more familiar Ito formalism

dMt = Mt(Z(xt), Ut dBt>xt

and

t t

Mt = expJ (Z(xs),us dBs> - ~ J IZ(xs)l2ds 0 0

(211)

(212)

(213)

(214)

Proposition 18 Suppose <pt : t 2:: 0 is a c2,1 section of B* such that

o<; 11at =~trace v2<t>t + V<pt(A) + V<p1(Z) + J*(<;t) (215)

and <l't is bounded uniformly on 0 ~ t ~ T for each T > 0 . Then with the

assumptions and notation above, if also the process with generator ~6 +A+ Z

is complete and if J is bounded above

<t>t<vo) = IEMt<t>o<vt)

for each x0 in M and v0 E Bx0

where vt : t 2:: 0 satisfies the covariant

equation along the paths of x1 : t 2:: 0

199363

(216)

Proof. First consider the case 8 = M x IR with p the projection and J = 0. This

is the classical theorem: for cp: M.,. IR which is c2 and bounded the Ito formula

for Mtcp<xt) shows Qtcp defined by Qtcp(x0 ) = IEMtcp<xt) is a minimal semigroup on

L00 with differential generator ~6 +A+ Z; there is a unique such semi-group - -so the change of probability to IP with IP = MTIP (on paths w restricted to 0::;

t ::; T) is a change to a probability measure and under IP- the process x1 : 0 ::;

t ::; T has generator ~ 6 + A + Z.

Now Proposition 1A extends with essentially the same proof to the case

where the process Cxt: t 2:: 0 has a drift. Applying this to (x1 : t 2:: 0 under the

probability IP- gives (215). II

Note:

(i) Under the completeness conditions (Mt : t 2:: 0 is a martingale.

(ii) we can allow A and Z to be time dependent provided their sum A+ Z is not.

C. A case which we will be particularly interested in is 8 = TM and J(v) = Ric(v,-)•. From the Weitzenbock formula (199) proved later (Chapter V, §3),

and elliptic regularity which shows that solutions to the heat equation for

forms are c2,1 (in fact C00 ) we have from Proposition 1A and Yau's result on

the stochastic completeness of M when M is complete with Ricci curvature

bounded below:

Theorem 1C. If M is complete with Ricci curvature bounded below then

any solution cpt : t 2:: 0 to the heat equation for 1-forms

acptlot = ~ 6cpt

with 97t uniformly bounded on compact intervals [O,T] of IRis given by

ct't<vo) = IEcpo(Vt)

where Cvt: t 2:: 0 satisfies the covariant equation along Brownian paths

Dvtlot = - ~ Ri c(vt,- )•. II

We will discuss the analogous situation for p-forms p > 1 later.

§2. The top of the spectrum of 6

A. For complete Mit is a standard result that 6 is essentially self-adjoint on

the space of C00 functions with compact support (as is trace v2 acting on

sections of 8* as in §1 and the de Rham-Hodge Laplacian on forms, [91]).

Since, for f of compact support,

200364

J fM=- f<Vf,Vf):sO, M M

b. is non-positive and so there is a semi-group induced on l2(M) by ~b. which

we will write e~tb.: t 2: 0). This semigroup restricted to L00 n l2 extends to

a contraction semi group on L 00 (M), e.g. see [86] p.209. By elliptic regularity

and the simplest case of Proposition 1A, this implies that

e~tb. f(x 0 ) = !Ef(xt) (217)

for f e L 00 n L2. We will not distinguish between b. and its (self adjoint)

closure.

There is the heat kernel Pt(x,y) fort> 0 and x,y e M. It satisfies

Pt(x,y) = lim i~

()()

D· Pt T (x,y) (218)

where D;i=t is an increasing sequence of bounded domains in M with smooth

boundaries whose union is M and where Pt0

i(x,y) denotes the heat kernel in Di

with Dirichlet boundary conditions. Equation (218) holds because the

corresponding result holds for the transition probabilities of Brownian motion

on M and the Brownian motions in Di killed on the boundary.

For an incomplete manifold (218) can be taken as the definition of Pt(x,y),

each Di having compact closure.

B. Si nee b. is a negative operator :>-. 0 := sup :A. e Spec b. :s 0. When M is

compact or has finite volume :>-. 0 = 0 since the constants lie in L.2. There are

various characterizations of :A. 0 (M) e.g. see [93]: in particular

:A. 0 (M) =- inf J IV<pl2 I J l<pl2: <pis C00 with compact support). M M

()()

Let D;h=t be an exhaustion of M by pre-compact domains with smooth

boundaries as before. The spectrum of the Laplacian with Dirichlet boundary

conditions for functions on Di is discrete. Let :>-. 0 (0;) be the first eigenvalue,

so :A. 0 (Di) < 0. Then

201365

A. 0 (Di) = -inf U0

. IV<pl2/ f0

.1<pl2: <pis coo with 1 1

compact support in Di

00 e.g. see (28]. Thus A. 0 (Di)i=l is increasing and

A. 0 (M) = lim A. 0 (o1). 1-+00

It is shown in [31], see also [93], that if p E o 1 and hi : Di -+ IR satisfies

. . . . n hl(p) = 1 and t.hl = A. 0 (Di)h1 then on any compact set in M the sequence (hl)i=l

has a uniformly convergent subsequence giving a limit h : M -+ IR which is

positive and satisfies t.h = A. 0 (M)h. A. smooth function h is a A.-harmonic

function if t.h = A.h. A basic result [93] is (for non-compact M):

There are positive A.-harmonic functions if and only if A.~ >-o(M) •

Note that for A. * >-o such functions cannot be in L2.

C. The Green's region consists of those A. with

a>

g>-(x,y) = ~ J e-~ A. t p1(x,y)dt < oo 0

for all x,y with x * y. From functional analysis if A. > >-. 0 then A. lies in the

Green's region. See [6], [93]. On the other hand if A. < A. 0 (M) then A. < A. 0 (D;)

for some Di. Writing A.i for A. 0 (Di) and t,i for the Dirichlet Laplacian for oi, if

A. were in the Green's region this would imply from

• 1"\it 1tAi • h1(x) = e-2"' e2 u h1(x)

that

t t

hi(x) = 1/t Jhi(x)ds = 1/t J e-~ >-.is (Jp0

i (x,y)hi(y)dy)ds 0 0 M S

t

:S 1/t J e -~ >-s<JD. Ps(x,y)h i (y)dy)ds 0 1

202366

t

= JD· (1/t J e -~A.s Ps(x,y)hi(y)ds)dy 1 0

-+ 0 as t -+ oo.

Thus [93],the Green's region consists of [A. 0 (M),oo) or (A. 0 (M),oo).

Given a positive A.-harmonic function h:M-+ IR (> O) we can h-transform

Brownian motion, and the heat semigroup Pt. For this define, for measurable

f : M-+ IR, the function phtf:M-+ IR by

pht f(x) = 1/h(x) e-A.t/2 Pt(hf)(x) (219)

when it exists. This gives a semigroup with differential generator Ah where

Ahf(x) = ~M(x) + (V logh(x), Vf(x)>x (220)

so the corresponding Markov process is Brownian motion with drift Vlogh; this

is the h-transformed Brownian motion on M. The fundamental solution is

given by

Pth(x,y) := 1/h(x) e-A.t/2 Pt(x,y)h(y) (221)

From general results about transience and the existence of Green's operators

in [6 ], or from [93], we see A. is in the Green's region if and only if the h­

transformed Brownian motion is transient.

D. The following result from [93] will be very useful, as will its method of

proof which comes from [46. See also [85]

Proposition 2P If A. belongs to the Green's region then for every x0 eM

and compact set K of M, if xt : t 2: 0 is Brownian motion from x0

1. -(A./ 2)t IP K) - 0 rrn e Xt E - • (222) t-+OO

Proof Choose a A.-harmonic function h : M-+ IR(> 0). Let (Yt : t 2: 0) be the h­

transformed Brownian motion from x0 • It is transient and so lim IE XK(Yt) = 0 t-+OO

i.e. lim pht(XK)(Xo) = 0 t-+OO

By definition of pht this means lim e-~A.t lEh(xt)XKCxt) = 0 t-+OO

which gives (222) since his bounded away from 0 on K.

203367

Corollary 20 ForK compact and (xt : t 2: 0 Brownian motion on M

lim 11t log IP(Xt E K :$ ~ A. 0M) (223) t-+OO

Proof By (222) if :>... > :>... 0 (M) then lim 11t log IPXt e K :$ ~ :>.... I I t-+00

§3. Bochner theorems for L2 harmon1c forms

A. There is a wide literature in differential geometry relating curvature

conditions on M to the existence of functions, forms or tensors, of particular

kinds e.g. see [14], of these Bochner's theorem has been particularly

important: the simplest case of it says that if the Ricci curvature of M is

positive definite and M is compact then there are no harmonic 1-forms. The

importance is because of Hodge's theorem which states, in particular, that the

dimension of the space of harmonic 1-forms is the 1st Betti number of M when

M is compact i.e. it is the dimension of the first cohomology group H1 (M;IR) of

M with real coefficients. This case of Bochner's theorem is very simple from

what we have done in Theorem 1C with equation (202). It is also almost

immediate functional analytically from Wei tzenbock's formula and the

following. Let q;' n_ (TM; T*M) be the space of C00 sections of the tensor

bundle over M whose fibres consists of R.(T xM; T x*M) for x in M, with

corresponding notation for other spaces of sections.

lemma 3A Let V* : q;' R.(TM; T*M) -+ q;'(T*M) be the formal adjoint of

V : q;' T*M-+ q;' n.(TM; T*M). Then for <p e q;' R.(TM;T*M),

(V*r.p)x =-trace (Vr.p)x =- ~=t Vr.p(ei)(ei)

where e 1, ... ,en is an orthonormal base for T xM·

(224)

Proof Given r.p e q;' ll(TM; T*M) and IV e q;'CT*M) there is the one-form

given by v-+ <r.px(v),I.Jix> for veT xM with corresponding vector field x-+

<r.px(-), IJ'x)•. By the divergence theorem

204368

0 = J div (<p_(-), tp_)• M

J trace V(<p_(-), tp_)• M

=J M

L (V<p(ei)(ei),tp) + (<p(ei), Vtp(e;). i

But ~(<p(ei), Vtp(ei)) = (<p,Vtp) and so (224) follows. I I 1

From Lemma 3A the Weitzenbock formula (199) can be written

t:.<p = -V*V<p- Ric(<p*,-) (225)

for a 1-form <p. Thus if <p has compact support and is smooth

(6<p,<p)L2 =- (V<p, V<p)L2 - J Ric (<p•, <p•) (226) M

from which Bochner's theorem for compact M with Ric(v,v) > 0 for all v * 0

follows. In fact it clearly extends to the case of non-compact M if we consider

only L2 forms <p (and are careful about the existence of V<p in L2 if t:.<p = 0).

See [39], for example, for generalities about L2 harmon1c forms etc.

B. The follow1ng is an 1mproved version of the L2 Bochner theorem, taken

from [46]. The case of strict equality can also be fairly easily obtained

analytically, using the method of domination of semi-groups a direct parallel

of the proof given here) as 1n [40] §4. For x eM set .Bk(x) = inf Ric(v,v): lvl =

1, vET xM.

Theorem 3B [46]. Assume M is complete with B.k(x) 2:: A. 0 (M) for all

x eM and also either:

(i) B.k (x) > A. 0 (M) for some x e M or

(ii) A. 0 (M) is in the Green's region.

Then there are no L2 harmonic 1-forms except 0.

Before giving the proof we need a few more facts about the Laplacian on

forms. It is usually deflned (although with the opposite sign) on smooth forms

by

!;, = -(d& + &d) (227)

where d is exterior differentiation and & is the L2 adjoint of d. These are

205369

discussed in more detail later, and the Weitzenoock formula proved. For the

moment simply observe that if tp is a sufficiently regular one-form then

(6tp,tp)L2 = -(dtp,dtp)L2 - (otp, otp\2 (228)

since dd = 0. Thus 6 is a negative operator. It is essentially self adjoint on

c; and so there is a naturally defined semigroup (e~t6: t > 0). By regularity

theory, if the Ricci curvature is bounded below, Theorem 1C identifies e~t61P with PtfP• when tp is in L2 n L00 • for Pttp(v 0 ) = IEtp(Vt) with the notation of

Theorem 1C. See [91a] and the discussion in [46].

When we wish to distinguish between the Laplacian on forms and on

functions we will use 61 and 6o, with pit and pot for the corresponding

probabilistically defined semigroup.

Proof of Theorem 38. Suppose there is a non-zero L2 harmonic 1-form tp 0 •

Choose a smooth Jl : M-+ 1R(2: 0) with support in some compact set K such that tp

:= Jl'Po is not identically zero. The space of L2 harmonic 1-forms is closed in

L2 (it is (tp e L2: et6tp = tp for all t > 0). Let H be the projection in L2 onto it.

Then Htp * 0 since

(Htp, 1Po>L2 = (fP,fPo>L2 = J J.l<fPo,fPo> > 0. M

By abstract operator theory e~t6tp-+ Htp in L2 as t-+ oo. A subsequence

therefore converges almost surely on M, say on some subset M0 of M. Choose

x0 E M0 • Set v0 = (Htp)•x E T x M. Then 0 0

(229)

Set C = inf Ric(x). For Dvt/ot = - ~ Ric(vt,-)• along Brownian paths, by

Theorem 1C and estimate (202), equation (229) gives

(230)

which implies C is not in the Green's region by Proposition 20. By assumption

206370

C :::: :>.. 0 (M). Therefore C = :>.. 0 (M) and :>.. 0 (M) is not in the Green's region i.e. (ii)

does not hold.

To contradict (i) assume it holds and take a :>.. 0 -harmonic function h: M-+

IR(> 0) for :>.. 0 = :>.. 0 (M) = C. Let z1 = xht• the h-transformed Brownian motion

starting at x0 • Its generator is ~6 + Vlogh. Since :>.. 0 is not in the Green's

region Zt: t:::: 0 is recurrent and hence complete and we can apply

the Girsanov theorem, Proposition 18, toP 1t(c.p) to get

e-~t6 c.p(vo) =IE Mt c.p(vt)

where vt: t:::: 0 satisfies Dv1;at =- ~ Ric(vt,-)• along the paths of

Czt: t :::: 0 and

t t

Mt = exp (- J <Vlogh(zs), UsodBs>- ~ J ( -6logh(zs)ds + IV log h(zs)!2)ds 0 0

t t

= exp -log h(zt) + 1ogh(x0 ) + ~ J 1Vlogh(zs)l2ds+ ~ J 6 logh(zs)ds 0 0

(240)

since 6logh = h-16h- 1Vloghl2 = :>.. 0 - IVloghF. Here u1 refers to the horizontal

lift of (Zt : 0 '$ t < oo.

Take a bounded open set V of M with ill (x) > o + C for x in V, some o > 0.

Let At= s E [O,t): Zs 1$ V and Bt = s E [O,t]: Zs e V. Then

t

exp - ~ J ill (zs)ds '$ exp -~ CIAtl- He + o)IBtD 0

'$ ex p (- ~ C t - ~ o IB t I

where IAtl• IBtl denote the Lebesgue measures of the random sets At• Bt.

Consequently

0 < lim IE M1c.p(vt) t-+OO

'$ lc.piL 00 h(x0 )(i nf h(x) : x e K )-1 lim IE exp (- ~ o!B 11)

since :>.. 0 = C. However IBtl-+ oo as t-+ oo almost surely by the recurrence of

zt: 0 '$ t < oo (e.g. see [5] proof of Lemma 1), so this is impossible. Thus (i)

207371

cannot hold either. I I

Corollary 38 If M is complete and has a non-trivial L2 harmonic one­

form then

A. 0 (M) 2: inf ...B.k(x) X

(241)

with strict inequality if either Ric is non-constant or A. 0 (M) is in the Green's

reason. II

This compares with Cheng's estimate for A. 0 when inf B..i.k.(x) < 0, [30]:

A. 0 (M) 2: 114 (n-1) inf x Bi.£x) (242)

and improves it for n > 5 given some non-trivial L2 harmonic 1-form

These results are discussed in relation to quotients of hyperbolic spaces

in [46].

Remark 38

(l) Corresponding results for p-forms can be proved in the same way given the

Weitzenbock formula for the Laplacian on p-forms (see below), and similarly

for the Dirac operator, [46]. The discussion in §1 shows how to formulate a

general theorem.

(ii) For compact manifolds Theorem 38 reduces to the classical Bochner

theorem. Note that the flat torus s1 X s1 has Ao(M) = .B.i£(x) = 0 for all X but

has harmonic 1-forms, e.g. d91 and d92 where (a1, a2) parametrize s1x s1 by

angle. Thus some additional conditions like (i) or (ii) are needed.

§4. de Rham cohomology, Hodge theory, and cohomology with

compact support.

A. Let AP be the space of C00 p-forms on M. (See Chapter V, §3.) Exterior

differentiation d gives a map

d: AP-+ AP+1

and the p-th de Rham cohomology group HP(M;IR) is defined by

p p+l HP(M;IR) = ker(d: A -+A )

Im(d: Ap-t-+ AP) (243)

It is a classical result that it is isomorphic to any of the standard cohomology

groups with real coefficients (e.g. simplicial or singular). The de Rham-Hodge

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Laplacian on p-forms, 6, or t:,P to be precise, is given by

t:.P = -(do + oct) (244)

where o is the formal adjoint of din the L2 sense. On the space Ab of p-forms

with compact support it is known to be essentially self-adjoint, e.g. see [91],

and so we can take its closure which will be self-adjoint. This will still be

written as t:,P. There is then the corresponding heat semigroup e~tt:.P acting

on the space L 2 AP of L 2 p-forms since t:.P is non-negative by the same

argument as for t:,1 ; see equation (228).

Let H = HP: L2AP-+ L2 AP be the projection onto the space of harmonic

p-forms. Then, as before, e~tt:.-+ H strongly on L2AP. For rp € L2AP set

00

Grp = J (e~tt:.- H)rp dt. 0

Then, leaving aside rigour for the moment,

00 00

t:.Grp = J t:.e~tt:.rp dt = J o/ot(e~tl::.rp)dt = Hrp- rp. 0 0

Thus we have the decomposition for rp e L2AP

rp = -t:.Gcp + Hcp (245)

From (244) we may believe the Hodge decomposition theorem, at least for

compact manifolds (when t:.P has discrete spectrum): any rp e AP has a

decomposition into three orthogonal summands

rp = Hcp + dcx + o j3 (246)

for ex e L2Ap-1 and j3 e L2Ap+1. In particular if dcp = 0 then cp = Hcp + dcx

since (cp, &J3\ 2 = (dcp,j3)L 2 = 0. Thus we have Hodge's theorem: every

cohomology class has a unique harmonic representative. In particular the p­

th Betti-number J3p

J3p := dim HP(M;IR) = dim (space of harmonic p-forms

when M is compact.

For non-compact manifolds the heat equation method outlined above was

used by Gaffney to get a version of Hodge's theorem [53]; see also [37], [38].

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B. The operators d and o can be restricted to the space A b to give the

cohomology groups HPK(M;IR) of M with compact support by the analogue of

(243). There is a natural inclusion

i p : HPK(M;IR) -+ HP(M;IR)

whose kernel has elements [<p]K represented by forms <p e A b with <p = dex

some ex in Ab. If <p' = dex' is also in Ab then [<p']K = [<p]K if and only if ex= ex'

outside of some compact set.

In the special case p = 1 this gives a linear surjection

E(M) d. -+

where E(M) is the quotient of the vector space of bounded ceo functions

f : M-+ IR with df e A 1 0 by the space A 0 0 of ceo functions with compact

support. The kernel of this map consists of [f] : f is constant and so we have

an exact sequence

The set of ends EndM of M is the projective limit of the inverse system

whose terms are the sets of connected components of M-K as K ranges over all

compact subsets of M, directed by inclusion. Thus an end is an indexed set EK

: K compact) such that EK is a component of M-K and if K c K' then EK• c EK.

They can be thought of as the components of Mat infinity and we will say that a

continuous path Cl : (O,eo)-+ M goes out to infinity through the end E or 'lim Cl(t) t-+OO

= E' if for each compact K there exists tK with Cl(t) e SK for t > tK.

There is a natural map j: S(M)-+ IREnd(M) given by j([f])(S) = limKf(x): x e

EK) which is clearly injective. Thus M is connected at infinity (i.e. has a

unique end) iff S(M) = IR.

C. A simple example is that of the cylinder M = s1 x IR. This has two ends, so

S(M) ~ IR ffi !R. If it is parametrized by (e,x) where e represents the angle,

210374

there is the one form de which determines the generator of H1(M;IR), which is

isomorphic to IR since M is homotopy equivalent to s1. Now [de] clearly does

not lie in the image of i 1 (otherwise it would have to be exact outside a

compact set i.e. equal to df , for some function f , on the complement of some

compact set). Therefore H1 0 (M;IR) ~ IR generated by [df] where f(x) = +1 for

x > 1 and f(x) = -1 for x < -1.

D. The relevance of these concepts to L2 harmonic form theory and Brownian

motion comes from the Hodge decomposition for cp e A b, [37], with dcp = 0,

which gives

cp = Hcp + do:

foro: e AP-1. From this we see that if there are no non-trivial L2

harmonic p- forms then the map

ip: HP 0 (M;IR)-+ HP(M;IR)

is identically zero. The point is that the latter is a topological condition

independent of the Riemannian metric (and in fact even of the differentiable

structure). From Corollary 38 we can now say that if i 1 is not identically zero

then :>.. 0 (M) ~ inf B.ll.(x); see [46] for some examples. This relationship X

between L2 harmonic forms and i 1 was exploited by Yau [10] for complete

manifolds with non-negative Ricci curvature. Using properties of such

manifolds (in particular the Gromoll-Cheeger splitting theorem) he was able to

give conditions for the vanishing of ker i 1 and hence for H10 (M;IR) itself rather

than just its image in H1(M;IR). Analogous results for B.k(x) > :>.. 0(M) are given

in [46], but different methods are needed, and additional conditions of 'bounded

geometry' appear to be needed for these methods to work. One such result is

described next.

§5. Brownian motion and the components of M at 1nf1nlty

A. It is shown in [ 46] that if M has bounded secti anal curvatures and a

positive injectivity radius (i.e. there exists r > 0 such that expx: T xM-+ M is a

diffeomorphism of the ball of radius r about 0 onto some open set in M for each

x eM) e.g. if M covers a compact manifold, then Ricx) > :>.. 0 (M) for all x implies

211375

that M is connected at infinity, and so by the previous discussion H10 (M;IR) = 0.

The proof uses some Green's function estimates by Ancona. Here we will

discuss another result from (46] which is very similar but has a slightly

different emphasis.

Theorem 58 (46]. Let M0 be a compact Riemannian manifold. Assume there

exist € > 0 with

J Ri cx)f(x)2dx > - J IVf(x)Fdx + €

~ ~ (247)

for all C00 functions on M0 with lfiL 2 = 1. Then every covering manifold M

of M0 is connected at infinity (i.e. has at most one end). Moreover H10 (M;IR)

= 0 and M, with covering Riemannian structure, has no non-trivial harmonic

1-forms in L2.

The condition on M0 is precisely the condition that the top of the

spectrum of ).-Ric is negative.

Proof: To show connectness at infinity take f : M -+ IR smooth and bounded

with df having compact support. It suffices to show that such a function f is

constant outside of any sufficiently large compact set.

Let u1, u2 be among the unbounded components of M-supp(df). For p : M

-+M 0 the covering map, take x0 eM and choose xi: 1 in u1 and yi: 1 in U2

with xi-+ oo and Yi -+ oo and p(x;) = x0 , p(y;) = x0 for each i. (We can assume M

non-compact of course.) Take a finite set of generators for the fundamental

group TT 1 (M 0 ,x 0 ) and let g1, ... ,gr denote them together with their inverses.

Choose smooth loops 'h•····lr at x0 , with each lj in the class gi and with tlj(s)l = 1 for all s and j e.g. the lj could be geodesics. Take a shortest path ex from xi

to xi+l· Then p o ex is a loop at x0 and we can write its homotopy class [p o ex]

as a product gil ••• gis for some s • Lift lis to a path in M starting from

xi and lift the other corresponding paths in turn to start where the previous

lift ended and so give a continuous piecewise c1 path from x1 to xi+1.

Do the same for x1 to y1 and Yi to Yi+l for each i. Let 6: (-oo,oo)-+ M be

the curve obtai ned from the union of these lifts: it is piecewise C 1 and

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satisfies lcrsl = 1 for all s where cr(s) is defined and there exists T e IR such

that o(s) e u 1 if s <-T and o(s) e u2 if s > T.

Let ll be the set of all open subsets of M on which p is injective. For

compact K in M let c(K) be the minimum number of elements in ll needed to

cover K, and suppose U1•···•Uc are in ll and cover K. Then

00 00

J XK (6(s))ds ::s ! J Xu.(O(s))ds . 1 J

-00 )= -00

Now the intersection of the curve o with Uj decomposes into portions P 1, ... ,P r

where each Pi consists of pieces which come from lifts of li· Since p is

injective on Uj it maps each Pi injectively into li· Therefore

Joo XU· (6(s))ds ::s t J00

(Xp.(6(s)))ds J 1=1 1

-oo -oo

where .t(li) is the length of li· Thus for any compact Kin M

Joo XK(O(s))ds ::s c(K) t .t(l1·). 1= 1

(248) -00

We can now show that J P:(df) exists for t ~ 0 and converges to zero as t-+ (J

oo. To do this take a smooth flow of diffeomorphisms on M0 of Brownian

motions, F0 t(-,w) : M0 -+ M0 for t > 0, w e Q, e.g. a gradient Brownian flow

as in Chapter III. This is possible because M0 is compact. It lifts to a

smooth Brownian flow of diffeomorphisms Ft(-,w): M-+ M, t ~ 0, w e Q. Set

K = supp (df). Then by (202) and Theorem 1C

00

1J P:(df) I ::s J IP 1t(df)l(os)ds (J -00

213

00

:i J -co

377

t

IE(!dfi(Ft(<'(s)))exp- J Bk..(Fr(d(s))dr)ds 0

co t

:i ldfiL 00 J IE(XK(Ft(d(s)))exp- J ~(Fr(<'(s))dr)ds -co 0

where K = supp (df).

(249)

To avoid worrying about regularity properties stemming from the possible

lack of smoothness of B.1£ choose a smooth map p 0 : M0 -+ !R with Ric(x) 2: p 0 (x)

for all x and such that condition (247) holds for .B.i£ replaced by p 0 • Let v

denote the top of the spectrum of 6.-p 0 on M0 • The revised condition (247)

implies that v < 0. From Perron-Frobenius theory (e.g. see

[86]) there is a strictly positive h0 : M0 -+ !R (> 0) with

6.h0 (x) - p 0 (x)h0 (x) = vh0 (x)

for x e M0 •

Let h = h0 o p : M-+ IR(> O) and p = p0 o p • There is a flow for the h0 -

transformed Brownian motion on M0 and a lift of it to a flow Fht, t 2: 0, say, on

M of h-transformed Brownian motions. By the Girsanov theorem using the

analogous computation as that which led to (240), from (249) we get

co

1J P:(df) I :i ldfiL 00 J (IE XK(Fht(<'s))e~vt h(<'s)/(h(Fht(<'s))) CJ -co

co

:i const. e~vt J IE XKt(<'s)ds -co

t

exp - J (B.k- p)(Fhr(<'s))dr)ds 0

where Kt is the random compact set (Fht)-1(K). However c(Kt) = c(K) since ll is

invariant under those diffeomorphisms of M which cover diffeomorphisms of

M0 • Therefore

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r

I J P; (df) I :S const. e~vt c(K) ~~(li) (J

-+ 0 as t -+ oo.

Thus

On the other hand, by the compactness of M0 , all the curvature

tensors and their covariant derivatives are bounded on M0 , so by Theorem 58

of Chapter III, pis b.(df) =pis d(M) = dPos M (or alternatively by [52],

pis 6 df = c,p1s df = -(d& + M)P1s df = dPos M). Therefore

R t

J (Pt df- df) = lim J d J P0 s(M)ds a("t)d't CJ R-+00 -R 0

t t

=lim J P0 s(b.f)(<'(R))ds- J P0 s(M)(<'(-R))ds R-+oo 0 0

= 0

by dominated convergence and the 'C 0 -property' of the semigroup P 0 t : t 2: 0.

This last property says that Pt(g)(x)-+ 0 as x -+ oo for each t whenever g is

continuous with g(x)-+ 0 as x-+ oo. It was shown by Yau to hold for complete

manifolds with Ricci curvature bounded below, e.g. see [iOO]. Alternatively it

follows rather easily from the existence of a Brownian flow of

diffeomorphisms [46].

Thus

0 =lim J P; df = J df =lim f(<'(R))- lim f(<'(-R)), t-+00 CJ CJ R-+oo R-+oo

and so f]Ui = f1U2, proving the first part of the theorem.

Next we observe that M has no non-trivial harmonic forms in L 2 by

arguing by contradiction as in the proof of Theorem 38 but using the h­

transform this time for h as above. The triviality of H10 (M;IR) follows from

the discussion in §4. I I

215379

CHAPTER V HEAT KERNELS: ELEMENTARY FORMULAE. INEQUALITIES. AND SHORT

TIME BEHAVIOUR

§1. The elementary formula for the heat kernel for functions.

A. We will be following [47] and [81] fairly closely in this section. For a

Riemannian manifold M and continuous V : M ~ IR, bounded above there is a

continuous map

(t,x,y) ~ Pt(x,y)

(IR > 0) X M X M ~ IR

such that the minimal semigroup (Pt: t ~ 0 for ~6 + V has

Ptf(x) = f M Pt(x,y)f(y)dy t > 0 (250)

for bounded measurable f. This is the fundamental solution. If

Di : i = 1 to oo is an increasing sequence of domains exhausting M, with

smooth boundaries, and if pti(x,y) denotes the fundamental solution to the

equation

oft/at= ~Mt + Vft

on Di with Dirichlet boundary conditions then

D· Pt(x,y) = lim Pt 1 (x,y)

i-+OO

(251)

(252)

and the right hand side is an increasing limit. This is clear from the

Feynman-Kac formula, or alternatively we can define Pt(x,y) by (252), with

compact Di, and then Pt by (250). In either case in order to obtain an

expression for Pt(x,y) it will be enough to find one for the fundamental

solutions on each Di, with Di compact, and then take the limit.

B. To obtain exact formulae for these fundamental solutions we will need

some rather strong conditions on the domains, and on M. However these

conditions will turn out to be irrelevant when the asymptotic behaviour of

pt(x,y) as t.!. 0 is being considered, at least for generic x andy and complete M.

To describe these conditions we need to look in slightly more detail at the

exponential map.

First suppose M is complete. For p EM let

216380

U(p) = v e T pM: d(exppv,p) = lvl

and let oU(p) be its boundary and U0 (p) its interior. The following facts can be

found in [16], [29], [63]: The image Cut(p) of oU(p) is a closed subset of M

known as the cut locus of p, moreover:

(a) U0 (p) is star shaped from the origin in T pM

(b) expp maps U0 (p) diffeomorphically onto the open subset M- Cut(p) of M.

Example 1: M = s1. Here the exponential map wraps T ps1::::: IR around s1 as a

covering map (it is locally a diffeomorphism), and Cut(p) is the point antipodal

to p.

Example 2. M = sn for n > 1. This is quite different from M = s1 since the

exponential map is no longer a local diffeomorphism: it maps the whole

sphere radius n to the antipodal point of p. Again Cut(p) is this antipodal point.

Example 3. Real projective space: M = IRIP(n). This is the quotient space of sn

under the equivalence relation x - y if x is antipodal to y. It is given the

differentiable structure and Riemannian metric which makes the projection p:

sn-+ IRIP(n) a Riemannian covering. If x e IRIP(n) corresponds to the North (and

therefore the South) pole of sn then Cut(x) is the image under p of the equator,

a copy of sn-1. Thus Cut(x) is a submani fold, isometric to IRIP(n-1), in IRIP(n).

It has co-dimension one and so will almost surely be hit by Brownian paths

from x in IRIP(n).

Example 4. M = Hn, hyperbolic space. In §4G of Chapter III we saw that there

are global exponential co-ordinates about a general point p. Thus Cut(p) = f!J.

Example 5. Complete manifolds with non-positive sectional curvatures

("Cartan-Hadamard manifolds"). The Cartan-Hadamard theorem e.g. [65], [79]

states that for such manifolds (e.g. M = s1) each exponential map expp: T pM-+

M is a covering map. In particular it is a local diffeomorphism. (To prove this

see Exercise 1A below.) It follows that if M is simply connected then expp is a

diffeomorphism and Cut(p) = f!J for each pin M.

When Cut(p) = f!J, so that there exists a global exponential chart about p, the

point p is said to be a pole of M. If so, M is diffeomorphic to IRn and so is

essentially IRn with a different metric. The images under expp: T pM-+ M of a

point v such that the derivative Tv expp of expp at v is singular is called a

conjugate point of p along the geodesic expp tv : 0 ~ t < oo, and v itself is

217381

said to be conjugate top in T pM·

Exercise 1A Show that the derivative of expp at v in the direction w is given

by

Tv expp(w) = J1

where Jt: 0 :S t :S 1 is a vector field along exp tv: 0 :S t :S 1 with J0 (0) = 0

and DJt/otlt=O = w. (Here Tv T pM is identified with T pM using the vector space

structure of T pM). Hint: look at the proof of Lemma 98 of Chapter III. Thus v

is conjugate top in T pM if and only if there is a non-trivial Jacobi field along

exp tv : 0 :S t :S 1 which vanishes at t = 0 and at t = 1. See [65], [79] for

example.

A basic result is that x e Cut(p) if and only if either xis the first conjugate

point to p along some geodesic from p, or there exist at least two minimizing

geodesics from p to x. For example when M = sn and p and x are antipodal then

both possibilities hold.

If r: M-+ IRis given by r(x) = d(x,p) then r is C00 on M - (Cut(p) u p)) since

there

r(x) = lexpp -1(x)lp (253)

B. Suppose now that Dis a domain in M- Cut(p) with D c W for W open with w compact and in M-Cut(p). We can use expp -1 to identify M-Cut(p) with the star­

shaped open set uo of T pM, and give uo the induced Riemannian metric. Then D

and Ware considered as sets in T pM· Using spherical polar coordinates in T pM

the Riemannian metric at a point v has the form

ds2 = dr2 + ~ g· ·(v)dai dai iJ=1 1)

(254)

where a1, ••• ,an-1 refer to coordinates on the sphere sn-1. Since the space of

Riemannian metrics on any manifold (and on sn-1 in particular) is a convex set

in a linear space, it is easy to first modify gij outside of W, if necessary so

that it extends to a metric on the whole of sn-1 for each sphere in T pM about p

which intersects W, and then modify this family of metrics (one for each

relevant radius lvl) outside of D and extend so that we obtain a Riemannian

metric on the whole ofT pM of the form

218

ds2 = dr2 + ~ h· ·(v)dcsi dcsi i,j=1 l)

382

(255)

which agrees with the original one on D and agrees with the standard Euclidean

one coming from<, >ponT pM outside of some compact set.

This gives T pM a Riemannian structure for which it is complete since the

geodesics from p are easily seen to be the straight lines from p, by the

distance minimizing characterization of geodesics, and the existence of all

geodesics from some point for all time is known to be equivalent to metric

completeness. The point p is now a pole and the curvature tensors are all C00

with compact support. Moreover the heat kernel for the Dirichlet problem in D

is unchanged since all these modifications took place outside of D. We can

therefore assume that M wasT pM with this metric.

c. Assuming the metric and manifold M has been changed in this way, and M

identified with T pM,

PtD(x,p) =lim J (2n)yn/2 PtD(x,y)exp- d(y,p)2/(2:A)ep(y)dy (256) A.Ul M

where dy refers to the Lebesgue measure of T pM, identified with M, using

< , >p• and ep is the volume element from the Riemannian metric (255): in

terms of our original metric it is given on T pM by

ep(v) = ldetM Tv exppl

and is known sometimes as Ruse's invariant. See [16] for more details about

it.

Thus

PtD(x,p) = lim P to f :>-. (x) A..l-0

where P1D : t ~ 0 is the Dirichlet semigroup for ~6 + V and

f). (x) = (2n:A)-nl2 exp -r(x)2/(2:A).

(257)

To evaluate PtD f). we will use the Girsanov theorem. Fix T > 0 and for :A ~ 0

let zAt : 0 :S t < T + :A be a Brownian motion on M from a point x0 of D with

time dependent drift z:As for z:As = VYAs with

y:As(X) =- r(x)2/(2(:A+ T -s))- ~ log 9p(X) 0 :S s < T +).

219383

Now iff: M-+ IRis smooth with f(x) = F(r(x)) for some smooth F: (O,oo)-+ IR,

by equation (4) for 6 in local coordinates, we have

M(x) = a2F/or2 + (<n-1)/(r(x)) + o/or (log ep)(x))oF/or (258)

In particular

6r = (n-1)/r + o/or log ep. (259)

By Ito's formula for r:At = r(zAt) we have

t t t

r:At = r(x0 ) + j dr(u:As dBs) + j o/or y:As(zAs)ds + ~ j 6r(zAs)ds 0 0 0

t t t

= r(x0 ) + j dr(u:As dBs)- J r:As/(:A+ T -s)ds + ~ J (n-1)/(rAs)ds (260) 0 0 0

where (u:As ; o :s s < T + :A) is the horizontal lift of (zAs : 0 :s s < t to the

frame bundle OM and (Bs: 0 :s s < oo) is a Brownian motion (which can be taken

to be independent of :A by taking the canonical construction of z:As from an

S.D.E. on OM).

Since ldrl = 1 and u:As(w) is an orthonormal frame the martingale term in

(260) is just a 1-dimensional Brownian motion, so r:A t : o :s t < T + :A

satisfies a stochastic differential equation which is essentially independent of

the manifold M. In fact from Ito's formula it satisfies essentially the same

equation (to be precise it is a weak solution of the same equation) as for the

radial distance (lxA s I: 0 :s s < T + :A where x:A s : 0 :s s :s T + :A is the

Euclidean Brownian bridge from x0 to 0 in IRn::::: T pM in timeT +:A, given by

s

x:As = x0 - sx 0 /(T +A)+ (s-T -:A) j d~tl(t-T -:A) (261) 0

where J3s: 0 :s s < oo) is Brownian motion on IR, e.g. see [59]. This Brownian

bridge is itself equal in law to

s-+ x0 + Bs- s (BT+:A + x0 ) (T+:A)-1 (262)

Thus r:As : 0 :s s < T + :A) is equal in law to lx:Asl: 0 :s s :s T + :A). In

particular it is non-explosive: as s t T +:A so it converges top (now identified

with the origin).

220384

Lemma 1C As A .1. 0 so (zA 5 : 0 s s s T converges in law to the process

zs : 0 s s s t which is sample continuous, agrees with z0 s for 0 s s < t

and has zT = p. Furthermore zs : 0 s s s T) has radial component r(zs) : 0

s s s t which has the same distributions as the radial component of the

Euclidean Brownian bridge in IRn starting from a point distance r(x 0 ) from 0

and ending at Din timeT.

Proof Let dxt = X(xt) o dBt + A(x 1)dt be a smooth stochastic differential

equation on M whose solutions are Brownian motions on M and, identifying M

with T pM and so with IRn, such that A has compact support and X(x) = o outside

some compact set. Then we can represent zA1 : 0 s t < T + A as the solution

to

(263)

Fix t 0 E (O,T). Then zAt: 0 s t s t 0 ) converges uniformly in probability

to z 0 t: 0 s t s t 0 . (Indeed we can choose versions so that it converges

almost surely since the coefficients of (260) have derivatives bounded

uniformly on [O,t 0 ].) Now suppose 0 < t < 1 and o > 0. Set t 1 = min t/3,

to/18 and choose o 1 > o such that

IP(r0 s < t1 for t-o 1 s ssT> 1-t1 > 1- t/3,

(which is possible by the continuity of r 0 s at s = T).

Take o2 > 0 such that for o < A < o2

IPIzA5 - zsl so for 0 s ssT -o1 2:: 1-£/3

and such that

IE[sup(rAs A 1: T -o 1 s ssT] s £1 + IE[supr0 s" 1: T -o 1 s ssT]

(which is possible because rA 5 -+ rOsin probability uniformly on [O,T]).

Then, for 0 < A < o/2

IE[suprAs" 1: T -o 1 s s s T] s 2£1 + (1-£1)£1 s 3£1

whence

IP[suprAs" 1: T -o 1 :s ssT> o/2) s 6£ 1/o s £/3

and so

IP(IzAs - Zs Is o for 0 s ssT 2:: 1-£/3- £/3- t/3 = 1-£.

This prove uniform convergence of zAs : 0 s s s T to zs : 0 s s s T) in

221385

probability.

If x0 = p, we should have been a bit more careful because of the

singularity of the distance function at p, but this is no problem si nee it is only

the convergence for t near T that causes any difficulties. I I

The process zt: 0 :$ t :$ T) will be called the semi-classical bridge from

x0 top in timeT. In [42] it was called the Brownian-Riemannian bridge but

this suggestion by K.D. Watling seems preferable. In particular it emphasizes

the fact that it will not in general coincide with Brownian motion from x0

conditioned to arrive at p at time T .

D. The following comes from [81], following earlier results with more

restrictive conditions in [42], [43].

Theorem 1D Suppose 1M is a Riemannian manifold which has a pole p in the

sense that its exponential map expp maps an open star-shaped region of T pM

diffeomorphically onto IM. Let M be some open subset of 1M (possible M = IM)

with p e M. Then for V : M -+ IR bounded above and coRtinuous, the

fundamental solution to the minimal semigroup for ~6 + V on M is given by

Pt(Xo,p) = (2nt)-nl2 ep(Xo)-~ e -d(Xo,p)21(2t)

t

IE[Xt<'t)exp J (~ 9p~(zs)6ep -~(z0 ) + V(Zs))ds)] (264) 0

where zs : 0 :$ s :$ t) is the semi-classical bridge in M from x0 top in

time t, defined up to its explosion time 't. In particular the expectation on

the right hand side of (264) is finite.

Remark: By the 'semi-classical bridge' here we mean a process which is a

Brownian motion with drift Z0 s: 0 :$ s :$ t), zo as before, in the interval 0 :$ s

< t A 't) where 't is its explosion time in M (so that if 't < tit either goes out to

infinity or leaves M as s 1' 't), and which is sample continuous with value p at

timet if t < 't.

. 00 Proof: Choose a nested sequence of domains D1)i=t with smooth boundaries,

such that Di is compact, both p and x0 lie in D1

, and M is the union of the Di. Let

pit(x0 ,p) be the Dirichlet fundamental solution for of lot= ~Mt + Vft in Di.

Let 't\ be the explosion time from D; of the Brownian motion with drift

222386

ZAs : 0 :S s < T + A) starting from x. While we obtain an expression for

pit(x 0 ,p) we can assume 1M modified outside of Di as in §18. Thus "t\ is the

first exit time from o1 of (zAs: 0 :S s :S T +A).

By (257) and the Gi rsanov theorem

T

piT(x 0 ,p) = lim IE[XT<"tA·) MAT fA(zAT) exp J V(zAs)ds] (265) A..J..O - 1 o

where

T

MAT= exp (J -(ZAs(zAs), uAs o dBs> 0

T

- ~ J (-div zAs(zAs) + IZAs(zAs!2)ds) 0

for (uAs : 0 :S s < T + A) the horizontal lift of (zAs: 0 :S s < T + A in OM from

some frame u0 at x0 ; see equation (212).

Since zAs = VYAs• writing e for ep:

MAT= exp (-YAT(zAT) + yAo(xo)

T

+ J (o/os (ZAs))(zAs) + ~ IZAs(zAs)l2 + ~ 6YAs(zAs))ds) 0

T

= e(zAT)~ e(x0 )-~ exp(~ J e~ (zAs) 6e-~ (zAs) ds 0

T

- ~ r(x 0 )2/(A+T) + r(yAT)2/(2A)- n/2 J ds/(A+T-s))ds) 0

since 1 _1 1 2

-6 ~ log e(x) = e2 (x)6e 2 (x) - IV log e2 (x)!

and -6r2 =- 2- 2(n-1) -2r o/or loge by (258).

223387

T

exp -i r(x0 )2/(:A+ T)) lEX T <"tA. exp I V eff (zA s)ds (266) 1 0

where 1 1

V eff(X) = V(x) + ~ 92 (x)~ e-2 (x) (267)

To treat this limit carefully let C be the space of continuous paths o: [O,T]

-+ IM+, where IM+ is the one point compactification of IM, with o(O) = x0 • Lets :

C-+ lR u oo be the first exit time from Di. Let JPA, IP be probabilities induced

on c by zAs : 0 ~ s ~ T) and zs : 0 ~ s ~ T. By Lemma 1C, JPA-+ IP narrowly

(='weakly'). Since Di is compact and has smooth boundary

T

o-+ XT <S (o) exp I Veff (O(s))ds 0

is bounded and Riemann integrable for IP in the sense that it is continuous

except on a set of IP-measure zero (this is because there is probabHity zero of

the path of a non-degenerate diffusion hitting oDi without leaving Di). It

follows, e.g. see [88] p. 375, that

T

p iT(x0 ,p) = e(x0 )-i(2nT)-n/2 exp -r(x0 )2/(2T) IEXT <"tiexp I Veff(Zs)ds 0

(268)

If we now let i -+ oo the left hand side of (268) converges and the term

under the expectation on the right hand side is positive and non-decreasing in i.

The theorem follows. II

Remark 10. The above proof shows that the upper bound on V was not essential

provided we know that Pt(x 0 ,p) exists and is given as the limit of the Dirichlet

heat kernels.

Corollary 10. (i) [ 47] For a complete manifold M, if x0 ~ Cut(p), then

t

p1(x 0,p)?:(2nt)-n/2 ep(x

0)-i e-d(xo,p)

2/(2 t)IEXt<"t exp Iv eff(Zs)ds

0

(269)

224388

where Veff is in (267) and "t is now the first exit time of the semi­

classical bridge from M- Cut(p).

Proof: Replace both ll'v1 and M in the theorem by M-Cut(p) respectively and

observe that Pt(x0 ,p) is greater than the corresponding value of the kernel for

M-Cut(p). //

Corollary 1D was used to get results about the limiting behaviour as

t J. 0 of the trace, f M Pt(x,x)dx, in [47]. It is especially useful for small t

since in the limit it becomes an equality as in Corollary 1D(iii) below. When

Cut(p) is codimension 2 it has capacity zero (the Brownian motion never hits it,

e.g. see [51]) and so fundamental solutions at (x0 ,p) for M and M-Cut(p) are the

same if x0 ~ Cut(p). Thus:

Corollaru 1D(jil c.f. [81]. If M is complete and Cut(p) has codimension 2

(or capacity zero more generally) then if x0 ~ Cut(p) there is equality in

(269). //

The following is a well known result with both analytical and

probabilistic proofs e.g. see [7], [17], 1[59a, [80], [84].

Corollary 10 (jjf) Suppose M is complete and x0 ~ Cut(p). Then as

t J. 0

Pt(x0

,p) = (2nt)-n/2 e -d(xo,p)2/(2t) ap(x 0 )-~(1 + o(t)) (270)

proof: First choose a compact domain D with smooth boundary in M which

contains the geodesic of shortest length from x0 to p. We need now quote the

result that as t J. 0

Pt(X0 ,p) = p0 t(x 0 ,p)(1 + O(tk)) (271)

for k = 1,2,... . For this see [7], [80], or [33] when x0 = p, (the O(tk) can be

replaced by O(exp(-&/t)) for some & > 0). Thus we need only examine the

behaviour of PDt(x 0 ,p) as t J. o. Choosing D with D inside M-Cut(p) we can

therefore modify M outside of D as in §18, so that p is a pole and it is flat

outside of a compact set, and also we modify V outside D to give it compact

support. Following this we can use (271) in the reverse direction and consider

Pt(x0 ,p) for the modified M. For this we have

t 1 2 ( ) f V eff<zs)ds

Pt<xo,p) = (2nt)-n/2 ap(Xo)-:z e-d(xo,p) I 2t IE e o

and so the result follows since Veff is now bounded on M. II

225389

The following corollary was noted in [42], [43]. It can be compared with

the trace formula and asymptotics in [28], [32], [94) for example. When M is

complete and p has no conjugate points then expp: T pM-+ M is a covering map

and if T pM is given the induced metric to make it a Riemannian manifold, M0 ,

say, the origin 0 is a pole. If x0 E M there are at most countably many points

x'¥ 0 in M0 with expp(x'¥ 0 ) = x0 , one for each geodesic "t from x0 top: we take 6

to be the geodesic t-+ expp tx6 0 in the reverse direction. For fixed t there is a

semi-classical bridge in M0 from each x60 to 0 in timet. Let z6s: 0 s sst

be its image in M under expp· This will be called the "semi-classical bridge

from x0 top along the geodesic 6· in timet". There is also a corresponding

e"tp(x0 ) which is just e0 (x"t 0 ) evaluated in M0 •

Corollaru 1DC1yl. Suppose M is complete and the point p has no conjugate

points. Then

1 ft (V (z6s) + cx:6s)ds Pt(x

0,p) = (2nt)-n/2 :r e"t (x )-2 e-2(6)2/21 lEe o (272)

"t p 0

where the sum is over all geodesics 6 from x0 to p,with 1(6) the length of

6· and z"ts: 0 s sst the corresponding semi-classical bridge; also cx:6s: 1 1

o s sst is ~e 0 (x6s)2 t.e 0 -2(x6s) where e 0 is Ruse's invariant in TpM

from 0 computed using its induced metric and x6 s is the semi-classical

bridge in T pM from xl 0 top in timet.

Proof. Let U be a sufficiently small open neighbourhood of x0 so that its

inverse image under expp consists of open neighbourhoods u6 of xl 0 in T pM·

Let f be the characteristic function of U and f6 that of Ul. Then for the

semigroups Pt : t 2:: 0 and Pt: t 2:: 0 for ~t. + V and ~t. + V o expp on M and

T pM respectively we see

Ptf(p) = :r6

Ptfl(p) (273)

by the Feynman-Kac formula since Brownian motion on T pM from 0 covers

Brownian motion from p in M. Because ul is mapped isometrically to U it has

the same volume as U and so we can let U be a ball radius £ about x0 and let£ J.

0 to obtain

Pt(P,Xo) = :r 1

Pt(p,x l o)

226390

in the obvious notation. However Pt(p,x0 ) = Pt(x0 ,p) and similarly for Pt(p,x'¥0 ),

and so the corollary follows from the theorem. I I

Note: The Cartan-Hadamard theorem assures us that the hypotheses on p and M

in Corollary 1D(iv) are always true when M is a complete manifold with all

sectional curvatures non-positive.

Example 10 (i) [92], [93]. The simplest non-trivial example of a manifold

with a pole is n-dimensional hyperbolic space Hn. From §4G of Chapter III,

equation (181), we see that

ep(x0 ) =(sinh r (x 0 )1(r(x 0 )))n-1 (274)

from which, using (258), we have

When n = 3 and V = 0 we can deduce the well known formula for the heat kernel

of H3:

Pt(x,y) = (2nt)-312 e-tl2 e-d(x,y)21(2t) d(x,y)l(sinhd(x,y)) (276)

with corresponding exact formulae for non-simply connected 3-manifolds of

constant negative curvature obained by using Corollary 1D(iv). The heat kernel

for the hyperbolic plane H2 is computed analytically in [28]. For a recurrence

relation between the kernels for hyperbolic spaces of different dimensions

see [28), with [35) for more details.

Example 10(111) [81], [83], [49]. ForM = sn-1 note that if p is the North pole,

say, in polar coordinates (r,<') in IRn-1 (so <l e sn-2) the exponential map is

essentially the map (r,<l)-+ (cos r, (sin r,<l)) e IR x IRn-1. In particular it maps

the sphere about 0 radius r to an embedding in IRn onto an isometric copy of the

sphere in IRn-2 radius sin r. Thus the metric in normal polar coordinates is

ds2 = dr2 + (sinr) (standard metric of sn-2).

Thus

ep(x0 ) = (sin r(x0 )1(r(x0 ) )n-1 (277)

and

1 1 1 1 ~e2(x0 ) t.e-2(x0 ) = 8 (n-1)2 + 8 (n-1)(n-3)(11r2- 11sin2r) (278)

for r = r(x0 ).

227391

Again n = 3 is an especially nice case. Since Cut(p) has co-dimension 2 we

can use Corollary 1D(ii) to get, for V = 0, if xis not antipodal toy

Pt(x,y) = (2nt)-n/2 (r/sin r) e!t e-r2/(2t) IP t < "t

where r = d(x,y) and where "t is the first hitting time of Cut(y) by the semi­

classical bridge from x to y in time t. However as we saw in Lemma 1C the

radial distributions of this bridge are the same as those of a Brownian bridge

Cl3s : 0 :s s :s t, say, in IR3 from a point distance r from 0, to 0, in time t.

Thus, [81],

Pt(x,y) = (2nt)-n/2 (r/sin r) e~t e-r2

12t IPsupO:ss:so'J3sl :s n (279)

This formula is discussed in [83]. In [43] it is used to obtain the exact

formula, for x,y not antipodal,

Pt(x,y) = (2nt)-3/2 e!t L .e(6)/(sin.e('¥)) e-.e('¥)2/(2t) "t

(280)

where the sum is over all geodesics 1 from x toy and .e('¥) is the length of 6·

Note the similarity here with the case of s1, or the s1tuation in Corollary

1D(iv). However in this case we no longer have a sum of positive terms. This

formula is a special case of a general formula for compact Lie groups, [50],

proved using harmonic analysis on such groups.

§2. General remarks about the elementary formula method and its

extens1 ons.

A. The way we were able to get a tractable formula for the heat kernel in the

last section depended on a suitable choice of drift Z:>'s: 0 :s s < T +A for

which there were convenient cancellations after the use of the Girsanov

theorem, and which gave processes with a very nice radial behaviour. In fact

the choice of ZAs came from a general philosophy outlined in [42], which is

explained below. However first it should be noted that there are various ways

of getting 'bridges' from x0 to p e.g. see [ 17]. The standard one is Brownian

motion from x0 conditioned to be at p at time T. This can be described as the

h-transform of (space time) Brownian motion where hs(x) = p0 T -s(p,x) for

p0 t(x,y) the fundamental solution for M when V= 0. This is used in [36], [80]

and [ 104]; it is Brownian motion with drift Vlog PT -s<P.- ). Writing it as xs :

228392

0 s ssT it is immediate from the Feynman-Kac formula that the kernel with

a potential Vis given by

T

PT(x0 ,p) = p0 T(x0 ,p) IE exp J V(xs)ds. 0

It has the advantage over the semi-classical bridge of symmetry in x0 , p i.e.

the reversed time bridge is the bridge from p to x0 • However the radial

behaviour will not be so pleasant in general.

B. Let Pt : t ~ 0) be the heat semigroup associated to ~6 + V on M. Suppose

g0 (x) = exp(- S0 (x)) ·f 0 (x) where S0 also are smooth functions on M with S0

bounded below and with f 0 of compact support. The drift terms for the semi­

classical bridge arose, [42], from seeking a nice expression for Pt g0 which

would ex hi bit its behaviour as A .1. o when S0 = A -1 R0 some R0 • Here is a

brief description. Assume for simplicity that M is complete and V, VS 0 and

the curvature tensor are all bounded on M.

First we associate to g0 the classical mechanical system with trajectories

g?t(a): t ~ O) for each a in M, satisfying

D/ot <i>t(a) = o (281)

with g? 0 (a) =a and <i>0 (a) = VS 0 (a). Under our assumptions it is shown in [42a]

that there exists T > 0 such that g? 1(a); t ~ 0 is defined for all 0 s t s T and

determines a diffeomorphism

g?t : M -+ M.

This is a 'no caustics assumption.

For this T we can define the Hamiltonian-Jacobi principle function.

s: [O,T] X M-+ IR

given by

I

S(t,a) = s0 (g?t-1(a)) + ~ J id>s o g?t-1(a)l2 ds (282) 0

There is then the following standard lemma, as in [42a]:

Lemma 28

(i) <i>t(a) = VS(g?t(a),t) OstsT, aEM (283)

(ii) S satisfies the Hamilton-Jacobi equation

229

~IVS(x,t)12 + as;at (x,t) = o with S(x,O) = S0 (x)

(iii) Define cp: N x [O,T]-+ IR by

cp(x,t) = ldet T x ~- 1 tl

393

O:St:ST (284)

(using the Riemannian metric of M). Then cp satisfies the continuity equation

acp;at (x,t) + div(cp(x,t) VS(x,t)) = o (285)

Proof:

t

VS(x,t) = (T~-1 1)* VS0(~-1 1 (x)) + J D/os T(~s o ~ 1 -1)* (<i>s o ~-1 1(x)ds. 0

Integrate by parts to obtain VS(x,t) = <i>t o ~t-1(x)

yielding (i). Also

t

a ;at S(x, t) = dS 0 ( ~-1t(X)) +~ l<i>to~t -1(x)l2+ J<D ;as T ~ s<<i>t -1(x) ), <i>so~t -1(x))ds. 0

Integrate by parts again and use (i) together with the identity

T~t o ¢-1t + <i>t o ~t-1 = 0 (286)

(which comes from differentiating ~to ~ 1 -1 = 1d) to obtain (ii).

For (iii) take any C00 function f: M-+ IR with compact support. Integrating

by parts

JM div(cp(-,t)VS(-,t))(x)f(x)dx

=- f M cp(x,t)(VS(x,t), Vf(x))dx

= - f M (VS(~t(X),t),Vf(~t(X))dx

=- f M df(<i>t(X))dx (by (i))

=- d/dt f M f(~t(X))dx = -d/dt f M cp(x,t)f(x)dx =-f M CJ/)t cp(x,t)f(x)dx

giving (iii). //

Now run the classical mechanical flow backwards. Take tin (O,T] and set

es(a) = ~t-s<~-1t(a)) 0 :S s :S t, a eM.

Then

a;as es(a) =- VSt-s<es(a)).

Let CYt: t ~ 0) be a Brownian motion on M from x0 with time dependent drift

(VY sCx): 0 :S s :S t, x eM) for Y s<x) = - S(a,t-s). We can think of it as (8s(x 0 ):

230

0 !0 s !0 t perturbed by white noise.

Formula A c.f. [42] For 0 !0 t !0 T

t

394

Pt g0 (x 0 ) = exp (- S(x 0 ,t) lE [exp(J V(ys) - ~ l::.St-s<Ys)ds f o<Yt)] (287) 0

Proof:

t

Ptg 0 (x 0 ) = lE exp J V(xs)ds- S0 (Xt)f 0 (xt). 0

Apply Girsanov's formula to obtain an expectation with respect to Ys: 0 !0 s

s t. The exponential martingale which comes in is

t

expY 0 (x0 )- Yt<Yt) + J (o/os Ys<Ys) + ~ IVYs<Ys)l2 + ~ t:.Y 5 (Ys))ds 0

t

i.e. exp S0 (yt)- S(x0,t)- ~ J t:.S(ys,t-s)ds 0

using the Hamilton-Jacobi equation. I I

This method can be modified in various ways. To obtain information about

the limiting behaviour of pXt gX 0 as X J. 0 where gX 0 is as g0 but with S0

replaced by X -1 S0 and where pXt refers to the semi group generated by ~Xt:. +

XV one proceeds in essentially the same way and Formula A gives the 'W.K.B'

approximation. However in this case, and for us, a slight modification gives a

more useful formula, [ 42a]:

Formula B

t

P tg0 (x0 )= .f <9t(x0 )exp -S(x 0 , t)lE[exp~ J<p(Zs, t-s )-~ t:.<p~ (zs, t-s )ds f 0 (zt)J (288) 0

where zs: 0 !0 s !0 t is Brownian motion on M from x0 with drift

(VY s(X) ; 0 :5 s :5 t, X E M) for

Y s(a) = - S(a,t-s) + ~ log <p(a,t-s),

assuming this process is complete.

Proof. From the continuity equation

231395

6S(x,t) = - olot log cp (x,t) - <V log cp(x,t), VS(x,t)>

so that

6S(®s(a),t-s) = oloS log !p (®s(a), t-s).

If we use this as well as the Hamilton-Jacobi equation after the Girsanov

transformation the formula follows. I I

The reason for introducing these formulae (which have many variations)

here is that to obtain the elementary formula for the heat kernel we needed the

case S0 (x) = d(x,p)21(2X). For this, given that p is a pole, we have ~ 1 (x) =

(X+ t)X -1x in normal coordinates about p and St(X) = ~ d(x,p)21(X + t). Then

'Pt(x) = (X/(X+t))n ep(x)-1 ep(XI(X+t)x)

in normal coordinates. To obtain the 'elementary formula' we could have used

the process zs = zXs of Formula Bas in [42]. However the actual process zXs

we used is easier to handle and gives the same limiting process, the semi­

classical bridge, as X .1. 0.

C. This very simple approach to the study of asymptotic behaviour seems to

have wide applicability, applying to both the Schrodinger and the heat

equations. In the former there is no Girsanov theorem, but this is made up for

by unitarity of the semigroup, and the use of a transformation of semigroups:

essentially an h-transform. This semigroup approach was worked out by

Watling [9Ba] to deal with both types of equation almost simultaneously. He

showed how it could be used to obtain full asymptotic expansions with exact

remainders. This was extended by Ndumu [82], and here we give a brief

description of how to get the asymptotics of the heat kernel p1(x 0 ,p) for ~ 6 +

v. Assume that p is a pole forM, with M complete and Euclidean outside some

compact region for simplicity, and that Vis bounded and smooth. Consider

qt(x,p) = (2TTt)-nl2 ep(x)-~ exp (- d(x,p)2/(2t)) (289)

The first observation is that as a function of x, writing e for ep it satisfies 1 1

(o/ot)f t<x) = ~ 6 f t (x) - ~ e:z (x) 6e-2 (x) f tCx) (291)

and moreover as t .1. 0 it converges .to the Dirac delta function at p. Next define

the 'semi-classical' evolution (Qp(t,s) : t 2: s > 0 on bounded measurable

functions by

Op(t,s)(f)(x) = q1(x,p)-1 Pt-s<qs(-,p)f)(x) (292)

232396

where Pt: t ~ 0 is the semigroup for ~ 6 + V. Another (this time standard)

computation yields for f smooth and with compact support

(o/ot) Op( t,s )( f)(x) = 0 6 + V log qt( -,p) + V eff )Oy( t,s )( f)(x) 1 1

where Veff(x) = V(x) + ~ e:z (x) 6e-:z (x) as usual. Consequently, now by a

Feynman-Kac formula rather than a Girsanov theorem, fort> s ~ 0

t

Op( t, t-s) ( f)(x 0 ) = IE[ exp d V eff(Zr )dr f(zs)] 0

(293)

where zs : 0 :::; s :::; t is the semi-classical bridge from x0 to p in time t. Letting s i t we obtain another proof of the 'elementary formula' (264).

To get the asymptotic expansion assume now that each pair of points x

and y in M can be joined by a unique geodesic. Let l(x,y) denote this path

parametrized proportionally to arc length so that l(x,y)(O) = x and l(x,y)(1)

= y. For f : M -+ IR and r ~ s ~ 0 define

F(r,s)(f): M-+ IR

by

F(r,s)(f)(x) = f(l(p,x)(s/r)) = f(s/r x)

in normal coordinates at p. Then, as for (292), for smooth f of ·compact

support

(o/os) [Qp(t,t-s)F(t-s,t-r)(f)J(x) = Op(t,t-s)~ [F(t-s,t-r)(f)J(x) (294)

where

y = ~!1- ~Vlog9.V+Veff· From (294) we have on integrating

Op(t,t-s) [F(t-s,t-r)(f)J(x)- Op(t,t)[F(t,t-r)(f)J(x)

s

= J Op(t,t-s 1) y[F(t-s1, t-r)(f)J(x)ds 1• 0

Setting r = s we get an expression for Op(t,t-s)(f) since F(t-s,t-s) = 1d. This

can be iterated arbitrarily many times by replacing f by g[F(t-s,t-r)(f)] and

substituting in the integrand, to yield a rather complicated expansion for

Op(t,t-s)(f) to arbitrarily many terms with a remainder consisting of a time

integral of various iterations of the operators. Knowledge of Op(t,t-s)(f)(x0 )

gives knowledge of Pt(x 0 ,p), c.f. (292). This way Watling's expansion [98a],

233397

[98] is obtained, see also [82], for N = 1,2, ....

Pt(x0 ,p) = (2nt)-n12ep(x 0)-~ exp( -d(x0 ,p )2 /(2t) )[ 1 + a 1 (x 0 ,p )t

+ ... +aN (x0 ,p)tN] + RN+l(x 0 ,p,t)tN+1 (295)

where

and

1

a1(x0 ,p) = J F(1, 1-r1)(Veff)(x)dr 1 0

aj<x0 ,p) = j i' ... t1

F(1,1-ri).F(1-ri'1-rj-1l ... 0 0 0

... ~F(1-r2,1-r1)(Veff)(x)dri ... dr 1

for 2 !i j !i N and

1 rl rN

RN+l(x 0 ,p,t) =lEd J ... J (gF(1-rN+l•1-rN) ... 0 0 0

trN+l

... ~F(1-r2, 1-rl)(Veff))(ZtrN+l) exp J Veff(Zs)ds)drN+1 ..• dr1].

0

As before this gives an exact expression when M has a pole and given some

additional bounds on its geometry [82], and furnishes an asymptotic expansion

when x0 $ Cut(p) for general complete M. It is easily modified to deal with the

fundamental solution to ~ 6 + A + V where A is a first order operator (i.e. a

vector field). Essentially the only differences are: (i) that 1

ep -2(x) is replaced by

1

ep -~(x) exp ( J <t(s), A(0(s))ds) 0

where 0 is the geodesic from x to p parametrized to take unit time, and (ii) ~6

is replaced by ~6 +A throughout; see [98].

234398

§3. The fermionic calculus for d1fferential forms, and the Weitzenbocl<

formula

A. The use of creation and annihilation operations for differential forms was

exploited by Witten for his approach to Morse theory (99]. The notation is very

useful in stating, and proving, the Weitzenbock formula for the Laplacian on p­

forms, as described in [33]. For our purposes it will enable us to give an

'elementary formula' for the heat kernel for forms especially suited to the

'supersymmetric' approach to the Gauss-Bonnet-Chern theorem which is

discussed later. It would be difficult to improve on the exposition in [33] and

it will be followed closely, as in [48] on which these sections are based.

B. First let us fix some notation and recall some basic facts. If Vis a real

finite dimensional vector space the space AP(V) of antisymmetric linear

maps cp: V x ••• x V-+ IR can be identified with the space of linear maps ll...(APV;

IR) i.e. (APV)*. If cp e AP(V) and ex e A 1(v) there is ex A cp e AP+ 1v given by

ex A cp(v 1, .•. ,vp+i) = !,1

(-1)i+1 ex(vj) cp(v 1, ... ,vj•···•vp+i) ]=1

(296)

where A indicates that the indicated term is omitted. This determines an

isomorphism of (APV)* with APV*, every element of the latter being

representible as a linear combination of terms of the form

ex1 A ••• A exP for exi E V*. If V has an inner product there is an induced inner

product on APV and APV* determined by

1 1 i . p (ex A ••• A cxP, J3 A ••• A J3P) = det [(ex, J3l)Ji,j=l

for exi, pi in V or V* respectively.

For such V, given cp e APV* and e e V define the "creation operator" a(e)*:

APV*-+ AP+1 V* by

a(e)*cp = e• A cp

where e• E V* is dual to e (it is most convenient to formulate it this way to

avoid a plethora of •·s later on). Let its adjoint be

a(e): AP+1y*-+ APV*.

Then

235399

a(e)(a111 ••• II aP+1) = ! 1

(-1)i+1ai(e)a111 ••. II ai II ••• II aP+1 (297) H

or as an antisymmetric linear map, for cp e AP+1(v)

a(e)cp(v 1, ••• ,vp) = cp(e,v 1, ••• ,vp) (297a)

There is the anti-commutation relation for

(a(e),a(f)*) := a(e)a(f)* + a(f)*a(e): APV*-+ APV*

withe, f e V:if cp e APV* then, from (297)

(a(e),a(f)*)cp = (e,f)cp (298)

C. A p-form cp on M gives an anti-symmetric p-linear map

fPx : T xM x ••• x T xM -+ IR

for each x e M. Thus we can consider fPx e AP T x*M and consider cp as a

section of the tensor bundle APT*M. Let AP denote the space of such sections

which are C00 and let A = ffip AP be the space of sections of A T*M := EBP APT*M. Supposing M is given a Riemannian structure (as we will from now

on) we can use the Riemannian measure and the inner products in each APT*xM

defined as above to obtain a space L2AP of l2 p-forms with inner product

(cp,\P)L 2 = f M <fPx• 'Px>dx.

Exterior differentiation d : A-+ A restricts to

d: AP-+ AP+1

for each p with

( dcp )x(v 1, ... ,vp+ 1) = f,\ -1)i+ 1ocp(x)( vi)( v 1, .•. ,v j, ... ,vp+ 1)

in local coordinates, where D is the Frechet derivative. Since the covariant

derivative agrees with the ordinary derivative at the centre of normal

coordinates for the Levi-Civita connection

(299)

From the definition we gave of covariant differentiation by lifting to OM it is

almost immediate that if cp, 'PeA then for v e TM

Vv(cp 11 ~P) = Vvcp 11 'P + cp 11 Vv~P (300)

236400

Let e1, ... ,en be an orthonormal basis forT xM then if <p e AP and v1, ... ,vp e

TxM

f..ca(e;)* Ve.<tJ)x(v 1, ... ,vp) = 2,. et AVe. <p(v 1, ... ,vp) j=l ' J j=1 J

= 2:i 2:k(-1)k+1<ej,vk> Vei'P(v 1, •.• ,vk, ... ,vp)

= 2:k(-1)k+1 v vk <p (v1, ... ,vk, ... ,vp).

Thus from (299) the two types of differentiation are related by

(d<p)x = 2:nj=1 (a(ej)* Vei ~P)x

Since a(ej)*(<p A If') = a(ej)*(<p) A If' = (-1)P <p A (a(ej)*IJ!)

equations (300) and (301) immediately yield

d(<p A I')= d<p A If'+ (-1)P !p A dl'

Consequently if cx1, .•• ,o::P are 1-forms

1 - "+1 < '+1 d(o:: " •.. " o::P)- 2:(-1)1 0::1 A ••• " do::l A o::J A ••. A o::P

(301)

when <p e AP

(302)

(303)

Let d* be the formal L 2 adjoint of d so d*: A-+ A restricting to

d*: AP+1-+ AP for each p. Thus d* = o in the notation of Chapter IV.

We already know, by (39) and (40), that the formal adjoint of V acting on

functions is minus the divergence: for a vector field A

V* A(x) = -di v A(x) = - Lj <V ejA(x),ej>·

Thus on 1-forms o::

(d*o::)x = - 2:j a(ej) V ejo::.

From (303) it follows that for <peA the same formula holds:

(d*~P)x =- Lj a(ei) V ej <p (304)

D. Let A be a section of IL(TM;TM), so for each x we have Ax:T xM-+ T xM. It

has adjoint

A* x : T x *M -+ T x *M

and can operate on A by

(A" (o::1 A •.. " o::P))x = -t o::1x" ••• "A*x(o::ix)" •.. A o::Px J=l

Observe that

A"x = -2:j,k A *kl a(ek)* a(e1)

(305)

(306)

237401

where A*kl"' <el, Aek>·

For vector fields A, B the curvature tensor R determines R(A(x), B(x)) : T xM ~

T xM for each x e M, which is skew symmetric. Recall from equation (83) that

if Vis another vector field

R(A(x), B(x))(V(x)) = ([V A• V 8 ]- V[A,B])(V)(x).

It is therefore immediate from (300) that as operators on A

[V A• V 8 ]- V[A,B] = R(A(·), B(·))" (307)

Recall that the (de-Rham-Hodge) Laplace operator 6: A-+ A is defined by

6 =- (dd* + d*d)

using the sign which makes it negative definite since d2 = 0 : see equation

(228).

Set Rijkl = R(ei,ej,ek,e 1) = <R(e;,ej)el,ek>• see (86). Note the sign difference

from [33], [88].

Proposition 30 (Weitzenbock formula)

6 = trace v2 - w (310)

where W, the Wettzenbock term, is the zero order operator given at x by

Wx = -L . . k 1

RiJ.kl a(e;)* a(el·)a(ek)* a(e1) (311) 1,), '

Proof Take normal coordinates at x, and using e1, ... ,en as a basis for T xM,

take E1, ... ,En to be vector fields on M which are C00 with compact support and

agree with the Gram-Schmidt orthonormalization of the fields a;ax1, .•. ,a;axn

at each point near x. Then for y near x, E1(y), ••• ,En(Y) forms an orthonormal

base for T yM· Moreover the covariant derivative of each ei vanishes at x.

Write a(j) for a(Ej( ·)) acting on forms by ( a(j)cp )x = a(Ej(x) )cpx, etc. By (30 1) and

(304) if cp e AP, summing over repeated suffices and working near x

dd*cp =- a(j)* vi a(k) Vkcp

so

Also

d*dcp = -a(k) V k a(j)* Vi cp

so

(d*dcp)x = -a(ek)a(ej)* (Vk Vjcp)x·

Thus

238402

(t:.cp)x = a(ei)*, a(ek) (V'Vkcp)x + a(ek)a(ei)* ([Vk,Vj]cp)x

=trace (V2cp)x + a(ek)a(ei)* R(ek,ejf

by (307) since [Ek,Ej] = 0 near x. Thus (310) holds with

W =- a(ei)a(ej)* R(ei,ejt (312)

= -a(ei)a(ej)* <R(ei,ej)ek,e1)a(ek)*a(el)

by (306), which agrees with (311) since a(ei), a(ej)*) = oij and R(ei,e;)

= o. II

Let t:.P : AP -+ AP denote the Laplacian acting on p-forms (i.e. the

restriction oft:.) and wP the corresponding Weitzenbock term.

The following special case of the Weitzenbock formula was used in

Chapter III:

Corollary 30 For a smooth 1-form cp

t:.1cp = ~ trace yr2cp- Ric(cp•,-),

Proof By (312) for veT xM

(W2cp)(v) = a(e;)a(ej)* R(e;,ej)*(cp)v

= a(ei) (ej •" (cp o R(ei,ej)(-))) (v)

= -<ej,v> cp(R(e;,ei)(e;))

= -<R(e;,v)(e;),cp•> = Ric(cp•,v). II

§4. An elementary formula for the heat k:ernel on forms

A. Assume M is complete. Then the de Rham-Hodge Laplacian on C00 forms

with compact supports is known to be essentially self-adjoint, [52], [91], [33],

and so determines a semigroup e~tt:. : t;:::: 0 on L2A which by elliptic

regularity has a kernel kt(x,y) which is C00 in t > 0, and x,y in M such that for

cp in L2A

(e~tt:. cp)x = JM kt (x,y)cpy dy (313)

with

kt(x,y): AT y*M-+ AT x*M.

By the same argument described for 1-forms before the proof of Theorem

3B of Chapter IV the Weitzenbock formula implies that if WP is bounded below

then e~tt:. determines a bounded map from L 00 Ap to L 00 AP and if cp e L2AP

239

is also bounded then

e~tt. r.p = Ptr.p

where

P1r.p(v 0 ) = IEr.p(vt)

403

for v0 E APT x0M with Vt(w) e APT Xt(w)M given by

(D/ot)(vt) = - ~ (WP )*(v1) Xt

(314)

(315)

along the paths of the Brownian motion xt: t <= O) from x0 • Here we must also

assume that M is stochastically complete (e.g. that Ric is bounded below on M).

Here (WPx)* is the dual of WPx: (APT xM)*-+ (APT xM)*.

B. To obtain a formula for the kernel like that of §1, assume that p is a pole

for M, work with q-forms (to avoid confusion!), take ex e Aq T p*M and choose

r.p e AQ with compact support and such that f.Pp =ex.

Fix "t > 0 and for A. > 0 define f.P>.. e Aq by

f.P>..,x = (2nA. "t)-n/2 exp -d(x,p)2 /(2>.. "t)r.px (316)

Observe that the l<erneli<Qt(x,y) for q-forms satisfies

I<Qt(x0 ,p )( o::)(v 0 ) = limA..l.O P tf.PA. (v0 ) (317)

In fact we will obtain a formula in a slightly different form to that in §1

and more adapted to describing the asymptotics as t J. 0. For this let Ht: t <=

O) be the semigroup et"tt./2 : t <= 0). Thus Ht = Pt"t· Let Xt : t <= 0 now have

generator ~ "tt.0 (where 6° is the Laplacian on functions) and let vt: t <= 0 be

defined by

(D/ot) (vt) =- ~ "t(Wxt)*(vt) (318)

Then

(319)

As in §1 apply the Girsanov theorem to obtain

P"t f.P>.. (v0 ) = e(zA. 1 )~ e(x 0)-~ (2TT"t(1+A.))-n/2 exp -d(x0 ,p)2 /(2(1+A.)"t)

1

IE[exp("t J ~ e~(zA.s) t. 0e-~(zA.s)ds)r.p(vA.1)] 0

(320)

where the processes (zA.s : 0 ~ s < 1 + A.) now have generators ~ "tt. 0 + vy>..s

for

240404

YAs(X) =- ~ d(x,p)2/(A+1-s)- ~'t log 9(x) (321)

and vAs: 0 :S s < 1 +A satisfies the analogue of (318) but along the paths of

z\ : 0 :S s < 1 + A.

To take the limit as A J. 0 we need to know that lim vA 1 exists, and to get a

sensible answer we would like this limit to be limsil v0 s. In particular the

latter should exist. For simplicity assume now that in normal coordinates

about p the manifold M is Euclidean outside some compact set.

Proposition 4A For each v0 e APT x M the limit limsil+A vAs exists almost 0

surely. Moreover in normal coordinates about p it exists in L2 uniformly in

0 :S A :S 1 and 0 < 't :S 1.

Proof From §4E of Chapter II and the stochastic version of equation (12), in

normal coordinates about p for 0 :S s :S t < 1 +A

t t

vAt- vAs=- J r(xAr)(odxAr)vAr- ~ 't J (Wq A )*(vAr)dr s s x r

Since WP is bounded lvArl is bounded independently of chance, and of

A, 't, r. Also !r(x)l < canst. lxl since it vanishes at the origin and has compact

support. Thus there exist constants c1,c2···· independent of s,t, A, 't and of

chance such that using the Euclidean inner product of our coordinates:

t

lvAt- vAsl :S c 11 J r(xAr)(J.l uAr o dBr)l s

t t

+ c2 J lxArl/(1 +A - r)dr + c3 J IV log e(xAr)ldr s s

(322)

where J.l2 = 't, uAr : 0 :S r <A + 1 is the horizontal lift of (xAr : 0 :S r < A+1,

and Br: 0 :S r < oo is a Brownian motion on !Rn. The radial component JxArJ is

now such that if pAr= J.l-1lxAr1 then pAr: 0 :S r :SA+ 1 has the same law as

the Euclidean Brownian bridge from J.l-1 x0 to 0 in time 1 +A, by the same

241405

argument as for the case Jl = 1. Thus it itself is equal in law to

r-+ lx0 + Jl Br- Jlr(B1+X + Jl-1x0 )/(1 +X)!

i.e. r-+ 1(1 +X- r)(1 + x)-1x0 + Jl(1+X)-1(1+X-r)B 1+X + Jl(Br-B 1+x)l

Thus the second term on the right hand side of (322) can be estimated by

t

c2 f I(1+X)-1x0 + J1(1 + X)-181+>.. + J1(1+X-r)-1(Br-B1+X)Idr s

t

~ c2 (t-s) + c3 IB1+XIIt-sl + c4 f IB1+X- Brl (1 +X- r)- 1dr s

= 0(11 + X - slex)

as s i 1 +X both almost surely, and in L2 independently of .Jl, X, for any

ex e (0,~). (using the pathwise Holder continuity of Brownian paths for the

almost sure case). The martingale term in (322) is equally tractable: since r is bounded and so is luX rl , it is a time changed Brownian motion with a

bounded time change. Thus, as s-+ 1 + X,

lvX1 - vXs! = 0(11 + X - slex)

for any ex e (O,~) both almost surely, and in L2 uniformly in Jl e (0,1] and X e

[0,1]. This gives the required result. II

Theorem 48 Let M be a complete manifold with pole p such that the

Weitzenbock term for q- forms, Wq, ts bounded below. Let (zs : 0 ~ s ~ 1

be the semi-classical bridge from x0 to p in time 1 with diffusion

constant "t: so it has ttme dependent generator ~"t6 + VY0 s, 0 ~ s < 1, for

yo given by (321). For v0 e .t\ qT x M let vt : 0 ~ t < 1) be the solution to 0

(318) along the paths of (zt : o ~ t < 1. Then

(i) v1 = limt-+oo Vt exists almost surely as an element of .t\GT pM;

(ii) the fundamental solution to the heat equation for q-forms is given by

kG"t(x0 ,p)(ex)(v0 ) = (2n"t)-n/2 ep(x 0)-~ exp( -d(x0 ,p)2/(2"t)

1

IE[exp~"t J ep(Zs)~.1ep -~(zs)ds ex(v 1)]. (323) 0

242406

Proof Part (i) follows from the previous proposition by progressively

modifying the metric of M to be Euclidean outside of larger and larger domains

as described in §18.

00 For (ii) take domains Dii=l as in the proof of Theorem 1D and observe

that k q 't ( x 0 , p) = lim k q • i 't ( x 0 , p) where k q • i 't refers to the fundamental i~oo

solution on Di with Dirichlet boundary conditions. To compute kq,i't we can

assume M is Euclidean outside of a compact set, in normal coordinates about p.

Use the canonical S.D.E. on OM with added drift to define xAs: 0 :S s < 1 +:A.

Then by standard results about S.D.E. with parameters as in [57] or [90] there

are versions of x:As : o :S s < t) and vAs : o :S s :S t for each t < 1 such that

v:As converges in L2 (in our coordinates) to Vs uniformly in s E [O,t]. By the

previous proposition it follows that v:A 1 ~ v1 in L 2 and so arguing as in the

proof of Theorem 1D

kq,i't(x 0 ,p) = (2rn)-nl2 e(x 0 )-~ exp -d(x 0 ,p)2/(2't)

1

IE[X; exp ~ 't J ehzs) 68-~(zs)ds cx(v 1)] 0

where Xi is the characteristic function of w E Q: Zs E Di for 0 :S s :S 1). Now

take the limit as i ~ oo: the result follows by dominated convergence since v 1

e L 00 and the exponential term is in L 1 by Theorem 1D. II

Example 48 (Hyperbolic space). For M = Hn the Weitzenbock term Wqx is just

multiplication by the constant -q(n-q). It follows that in (323)

V1 = exp nq(n-q)'t //V 0

where //v0 refers to the parallel translate of v0 along the paths of Zt: o :S t

:S 1. Thus for ex E A qT*pM and v0 E A qT x0

M

kq't(x0

,p)cx(v0

) = (2rn)-n/2 (r 1 si nhr )~ (n-1) e-r 2 /(2't )e~ q(n-q)'t

(324)

1 1 with 82 Czs)6e-2(zs) given by (275), and in particular equal to -1 when n=3.

243407

CHAPTER VI. THE GAUSS-BONNET -CHERN THEOREM

§1 Supertraces and the heat flow for forms

A. Let V be a real inner product space. If 8 e 0...(/\V*; /\V*), with gq its

restriction to /\qV*, define its supertrace, str B, by

str 8 = Lq (-l)q trace Bq.

For a fixed orthonormal basis e1, ... ,en of V we have the annihilation and

creation operators a( ei) and a( ei )* defined in §3 of the previous chapter. For I

= i l>····ik a naturally ordered subset of l, •.. ,n define

ai = a(ei1

) ••• a(eik) e 0...(/\V*; /\V*).

Followers of quantum probability will recognise these and recall, e.g. from

[78a] p. 221, that the collection (ai)* aJr,J forms a basis for D...(/\ V*;/\ V*).

Indeed there are the correct number of them, 22n, and they act transitively on

/\ V* since given ex e /\ V*, with ex * 0, it can be annihilated down to a non-zero

element of /\ 0 V* ~ R by a suitable aJ and any non-zero element of /\V* can be

created from a non-zero element of /\OV* by a suitable (al)*. Al1ernative1y

see [33] page 248.

A basic result, emphasized, and called the Berezin-Patodi formula in [33]

is:

Propositlon 1A. ForB= LI,J f3r,J<a1)* aJ,

str B = (-1)n .J31, ..• ,n), l, .•• ,n (325)

B. The de Rham cohomology groups, or vector spaces, Hq(M,IR) were defined

by (243) in §4A of Chapter IV. As described there, when M is compact, the

Hodge theorem shows that

dim Hq(M;IR) =dim ker t.q.

(The latter is finite by ellipticity of t.q). The Euler characteristic for

compact M can be defined by

X(M) = Lq (-1)q dim Hq(M;IR) = Lq(-1)q dim ker t.q.

Compactness of M implies that if Ptq = exp ~ tt.q acting on L2Aq then P1q

244408

is trace class fort> 0 and

trace Ptq = J M trace ktq (x,x) dx

e.g. see [33]. Define the supertrace of Pt = EBq Ptq acting on L2A by

str P1 = Lq (-l)q P1q t > o. There is the following remarkable fact due to McKean and Singer:

Proposit1on 18 ForM compact and all t > o StrPt = X(M). (326)

Proof Divide A into A+ EB A- where A+ = EBp even AP and A- = EBp odd

AP

respectively and for A e IR let n+(A) and n_(A) be the multiplicities of A as an

eigenvalue of 6+ and 6-, (the spectrum of 6 consists of a discrete set of

eigenvalues increasing to oo, see [28] for example).

Note that 6 = -(d + d*)2 since d2 = 0. Therefore if 6rp = AfP then 6(d+d*)rp

= A(d+d*)rp. Also (d+d*)rp * 0 unless drp = o and d*rp = 0 i.e. unless 6rp = 0.

Thus n+(A) = n_(A) for A * 0. However

Str Pt =LA (n+ (A)- n_(A))e-~tA Therefore

Str Pt = n+(O)- n_(O) = X(M)

as required. I I

From the proposition there is the following corollary

Corollaru 181) For all t > 0, when M is compact

X(M) = I M str kt(x,x)dx

§2. Proof of the Gauss-8onnet-Chern Theorem

(327)1 I

A. Suppose M is compact with even dimension, dim M = 2.g, say. The

G-8-C. theorem expresses X(M) as an integral over M of a certain function of

the curvature of M. We can do this by looking at (327) as t J. 0 using the results

of §1 and the 'elementary formula' (323) for the heat kernel. This is

essentially Patodi's proof as described in [33]. The difference from the

treatment in [33] is simply the use of the elementary formula. The Malliavin

calculus approach as in [59a] has the same structure but the cancellations take

place at the level of distributions on Wiener space i.e. before taking

245409

expectations. Related proofs of other classical index theorems are in [18],

[56], [59a], [68], [103], [104], and discussed ln [33]. The approach of [104]

fits in particularly well here, especially if it is simplified somewhat by using

semi-classical bridges: an important technique used in [104] is a rescaling of

Brownian bridges, and this seems to work equally well with semi-classical

bridges. The Atiyah-Singer index theorem describes the index of an elliptic

differential operator ~ (i.e. dim ker ~ - dim ker ~*) in terms of the

coefficients of ~. For the G-B-C. theorem ~ = d + d *:A+-+ A-. (Remember

t.rp = 0 iff d rp = d*rp = 0). The other 'classical index theorems' are for other

geometrically defined operators~. It turns out that the general result, for~ of

arbitrary order, follows from these special cases.

B. To examine the behaviour of str k,;(x,x) as 1: J. 0, or more generally str

k,;(x 0 ,p) when x0 • Cut(p) we can argue as Corollary 1D(iii) of Chapter IV and

assume that p is a pole for M and M is Euclidean outside some compact set in

normal coordinates about p. (Of course we have lost the compactness of M

after this modi fi cation.) Having done this, rewrite (323) as

kq,; (x 0 ,p) = (2n,;)-n/2 e(x0)-~ exp -d(x0 ,p )2/(2"t)

1

IE[exp !"t J Veff(Z("t) 5)ds~q"t] 0

where

~ q"t : A qT;M -+ A qT:0M

is given by

~q,;(cx)(v0 ) = cx(v 1).

(328)

using the notation of (323), but now we write Zs as 2(1:) 5 to make clear its

,;-dependence. Writing ~1: for ffiq ~q"t note that ~1: = ~1:,1 where ~,;,s:

0:$ s :$ 1) is the solution to

(D/os) ~1:,s =- ~ 1: ~1:,s Wz(,;)s (329)

along the paths of z(,;)s : 0 :$ s :$ 1) with ~1:,0 = 1d. This exists by

Propositi on 4A of the last chapter.

Now take an orthonormal base e 1(o), ... ,en(O) forT x0

M and let ej"t(s) =I Is 1:

246410

ei(O) be the parallel translate of ei(O) along 2(1:) from x0 to z(1:)s· Omitting

the superscript 1: unless it is needed, write

Rijkl(s) = (R(ei(s), ej(s)) e1(s), ek(s))

and set ais = a(ej(s)), a random annihilation operator. If we parallel translate

~1: s back to x0 and define H1: s E IL(A T:flM; AT: M) by • • v 0

then

for

~(s) = -i: Rijkl(s)(ai)* ai(ak)*a.e

where ai = ai 0 = a(ei(O)).

C. If we iterate the formula

1

H1:, t = Id - ~ 1: J H1:,s o~(s )ds 0

(330)

by substituting the corresponding expression for H1:,s back into the integrand

we obtain

H1:,i = Id + z1 + ••• Z,e + 0(1:h1)

where

1 St 8z

Z.e = (- ~ 1:).e J J ... J ~(s,e) o ... o ~(s 1 )ds 1 ... ds.e 0 0 0

for .e = 2,3, ... , and analogously for ..e = 1.

Thus by the Berezin-Patodi formula (325)

and

1 Sl Sz

str Z.g = (- ~ 1:)..e J J ... J z1:(s 1, ... ,s.g)ds 1 ... ds.g 0 0 0

where

(331)

(332)

247411

z't(s1 , ... ,SJ) = ( -1)1 ~ sgn(1t)sgn(cr) R1t(1)cr(1)1t(2)cr(2)(st) ...

··· R1t(n-1)cr(n-1)1t(n)cr(n)Cs1) where the sum is over all permutations 1t and cr of 1, ... ,n. (This is still a random

variable.) To get Z.t into the form in which we could read off its supertrace by the

Berezin-Patodi formula we have used the anti-commutation relations ai(aj)* + (aj)* ai

= oij and ai aj ;: -aj ai.

Now 4>1 ;: (I I 1 0 )* H-r,1 ;: H-r, 1 + [(/I 10 )* - IdlH-r, 1 . We claim that

str(l 11 0 )*-IdlH-r,1 = 0(-r), see §E below. Then from (328), (331) and (332) 1

strk-r(x0 ,p);: (-41t)-lep(x0 )-2 exp -d(x0 ,p)21(2-r)

1 lSl S2

lEiexp d 't J v eff (z('t)s)ds J J ... J z't(s1·····S.t)ds.t···ds.tl + O('t) (333) 0 0 0 0

Now let 't J, 0. Still with our assumptions on M we can choose versions of z('t)s : 0 s s

s 1 so that, almost surely, they converge uniformly on [0,1] to the geodesic from x0 to

p parametrized to take unit time. (Recall from (321) that z('t) had generator ~ 'tL\ 0 +

VYs0 with

Y sO(x);: - ~ d(x,p)21(1-s)- ~ 't log O(x).)

Correspondingly by Proposition 4A of Chapter V the horizontal lifts u('t)s:Osss1 will

converge to that of the geodesic, and z-r will converge to the corresponding non­

random term Z0 . In particular as 't J, 0 so str k-r(p,p)-+ E(p) where

E(p) = (41t)-.t11(l!) L sgn 1t sgn cr R1t(1)cr(1)1t(2)cr(2) (p) ...

R1t(n-1)cr(n-1)1t(n)cr(n)(p) (334)

for Rijkl(p);: <R(ej(O), ej(O))eJ(O), ek(O)>.

In particular the right hand side of (334) does not depend on the basis e1 (O), ... ,en(O) for

T pM. Note that, by equations (85) and (84 ),

R1t(l)cr(1)1t(2)cr(2) = -Rcr(1)cr(2)1t(2)1t(1)- Rcr(2)1t(1)1t(2)cr(1)

= R1t(1)1t(2)cr(1)cr(2) + R1t(1)cr(2)1t(2)cr(1) ··

-1

Thus E(p) = ~ Lsgn1tsgncr R1t(1)1t(2)cr(1)cr(2)(P) R1t(3)cr(3)1t(4)cr(4)(P) (41t) l!

... R1t(n-1)cr(n-1)1t(n)cr(n)(P)

248412

-l

= ··· = ~ L sgn1tsgncr R1t(1)1t(2)cr(1)cr(2)···RJt(n-1)1t(n)cr(n-1)cr(n) (41t) l!

which is the more standard expression for it.

D. From (327), Corollary 1B(i), we obtain:

Gauss-Bonnet-Chem Theorem. For a compact even dimensional manifold M

X(M) = f M E(x)dx where E is defined by (330).

The only thing we need to be careful about is the uniformity in x of the analogue

k-r(x,x) = k-rD(x,x) (1 + 0(-rk))

of (271) for suitable domains D which enabled us to replace M by a manifold with x as

a pole.

Special case: dim M = 2 Here

E(x) = (41t)-1 -R1221(x)- R2112 = (21t)-1 K(x)

where K is the Gauss curvature, see (87). Thus we have the classical Gauss-bonnet

theorem X(M) = (21t)-1 f M K(x)dx.

E. To see that str((/ 110 )*-Id)H't,l = O('t) consider the expansion of (I I 10 )*-Id in

terms of ai, aj* analogous to that for H-r,1 , using normal co-ordinates as in the proof

* of Proposition 4A. As an operator on AT x M , if us = u('t)s is the horizontal lift of 0

z('t)s , 0 ~ s ~ 1 , then

1 1

1

(I 110 )*-Id = J ~ (z('t)s) (odrl('t)s) aj* ak + 0

J Jr~\(z('t)sl)(odzjl('t)sl) r~z<z)('t)s2)(odzh('t)s2) .1ak1 a~2ak2 + .... 00

If we substitute dz(t)s = )lU('t)sodBs-(z('t')81(1-s) + hV log 9(z('t)s))ds,

and use the facts that r(x 0 ) = 0 and ).t -llz('t)sl, 0 ~ s ~ 1 , is a Bessel Bridge in IRn

from 0 to 0 it is easy to see that the p th term in this expansion is 0( 'tP) , p = 1 ,2, ....

Now multiply by the expansion of H't,1 and use the Berezin-Patodi formula (325).

See also [59a]. I I The expansion of (I 1 10 )*-Id plays a more important role in other index

theorems: [18], [59a], [104].

249413

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258

NOTATION INDEX

A , 400 A*, 298

ad(g), 298 a(e), a(e)*, 398 Cut(p), 380 d(-,-), 304 df, 284 dφ, 399 d*, 400 div, 307 D/t, 300 Exp, exp, 297, 301

ijg , 306 g ,ij 304

grad, 306 GLM, 297

nH , 315 H ,u 299 HTB, 298 K ,p 321

qtk (x, y), 402, 405

(TM;TM), 301 o(n), 313 O(n), 296 OM, 305

p (x, y),t 364

R(-,-), 320 Ric, 321

98 SO(n), 296 SOM, so(n), 312 TM,T M, T f,x x 283 T*M, 301 VTB, 297 α ,x 317

ijk, , 298

δ, 368 ∆, 307, 358

p, 368, 372 θ, 324

,t 331 p

Θ, 324

( ),0 364 , 335

Ω, 324 , 298

/ i,x 300, 303 <-,->x, 304

2, , 302, 306 |A|, 342

L

p,A 371

g,

, 382

M

s , 2

259

INDEX

A Action, 296–298, 305, 312, 313, 315, 342,

353, 354 Adjoint action, 296 Annihilation operator, 398, 407, 410 Atlas, 282, 283 B Bochner’s theorem, 359, 367, 368, 416 Bridge

Brownian, 383, 384, 391, 404, 409 semi-classical, 385, 388, 389, 391, 392,

395, 396, 405, 409 Brownian bridge, 383, 384, 391, 404, 409 Brownian motion

with drift, 308, 366, 385, 391 flow, 294 1-form, 311 on manifold, 280, 416l S.D.E., 308–310, 317, 383

C Cartan development, 312 Cartan-Hadamard theorem, 380, 390 Characteristic exponents, 331–359 Chart, 283, 284, 292, 295, 298, 300–302,

304, 306, 350, 380 Christoffel symbols, 298, 299, 305 Cohomology

with compact support, 371–374 de Rharn, 371

Complete manifold, 374, 378, 380, 387, 390, 405

Conjugate points, 380, 381, 389 Connection

affine, 300, 304, 308, 311, 320, 324, 327–329

form, 298, 305, 314, 323, 326, 327, 342 Continuity equation, 336, 338, 394 Cotangent bundle, 301 Covariant

derivative, 288, 299, 300, 302, 305, 306, 310, 317, 330, 339, 378, 401

equations, 335, 358, 362, 363 Covering space, Riemannian, 347, 348,

375, 380

Creation operator, 398, 407 Curvature

constant, 322, 326, 335, 347, 348, 351, 359

form, 323, 324, 326 Gaussian, 321, 322, 345 mean, 319 non-positive, 355, 380, 390 Ricci, 294, 321, 322, 345, 359, 363,

367, 369, 374, 378 sectional, 322, 374, 380, 390 tensor, 321, 325, 348, 378, 382, 392,

401 Cut locus, 380 D Diffeomorphism, 283, 284, 291–293, 296,

297, 301, 305, 312, 315, 324, 327, 331, 334, 341, 346, 350, 374, 377, 378, 380, 392

group, 291 Divergence, 307, 336, 367, 400

theorem, 307, 336, 367 Drift, 308, 362, 363, 366, 382, 385,

391–394, 406 E Elliptic, 306, 337, 341, 363, 364, 402, 409,

420 Embedding, 280, 283, 284, 286,

289–291, 293, 295, 305, 311, 317, 334, 338, 341, 352, 390

Ends, 315, 373, 375 Entropy, 337, 341 Euler characteristic, 345, 407 Explosion time, 280, 285, 287, 293, 308,

385 map, 293

Exponential map, 284, 297, 301, 316, 327, 379, 380, 385, 390

Exterior derivative, 323 F Feynman-Kac formula for forms, 280,

379, 392 Flow map, 288, 290, 293, 319

260

Frame linear, 297, 312 oriented, 312, 315, 353 orthonormal, 305, 308, 312, 313, 342,

383 Fundamental solution, 366, 379, 385, 388,

391, 397, 405 G Geodesic, 299–304, 309, 316, 327, 328,

349, 355, 356, 375, 380–382, 388, 389, 391, 396, 397, 411

Girsanov–Cameron–Martin formula, 362 Gradient Brownian systems, 318, 338 Greens region, 365, 366, 368–371 H Hamilton–Jacobi

equation, 392, 394, 395 function, 392

Harmonic forms, 367–371, 374, 378 Heat kernel, or density, 280, 364, 379–406 Hodge theorem, 367, 371–374, 407 Horizontal lift, 299–305, 310, 311, 319,

326, 335, 362, 383, 386, 404, 411, 412process, 299, 311, 362

Horocycles, 355 H-transform, 351, 366, 370, 377, 378, 391,

395 Hyperbolic space, 312–316, 326, 348, 352,

371, 380, 390, 406 I Injectivity radius, 374 Integration by parts, 338 Invariant measure, 331, 333, 336, 337, 341 Isometry, 313, 316, 342, 350, 353, 354 Ito formulae, 320 J Jacobi field, 327, 328, 330, 358, 381 L Laplace operator

eigenvalue, 364 on forms, 363, 368, 369 spectrum, 363, 364

Left invariant S.D.S., 316–319 vector field, 296, 313, 316, 326

Levi-Civita connection, 306, 307, 314, 315, 317, 320, 321, 324, 326, 327, 399

Lie algebra bracket, 296, 324 group, 296, 316–319, 355, 391

Lifetime, 287 Local

coordinates, 305–307, 383, 399 representatives, 300, 301, 306 trivializations, 297

Lorentz group, 314, 350, 353, 354 Lyapunov

exponents, 340, 348, 357 spectrums, 331–335, 342

M Manifold, 282–284 Maximal solution, 285–287, 292, 294,

308 Mean curvature, 319 Mean exponents, 334–339 Minimal semigroup, 363, 379, 385 Moment exponents, 281, 356–359 N Non-degenerate S.D.S., 304 Normal bundle, 284 Normal coordinates, 301, 303, 319, 350,

395, 396, 399, 401, 404, 406, 409, 412

O One form, 323, 367, 369, 371, 374

heat flow of, 367, 369 Orientable, 312, 342 Orthonormal frame bundle, 308, 312,

342 P Parallel translation, 280, 305, 310, 311,

319, 320, 335, 361 Path lifting property, 347 Pole, 312, 334, 380, 382, 385, 388–390,

395, 397, 403, 405, 409, 412 Principal

bundle, 288, 296–330 curvatures, 319, 321, 341 directions, 319

Projective space, 283, 284

261

R Ricci curvature, 294, 321, 322, 345, 359,

363, 367, 369, 374, 378 Riemannian

connection, 305, 306, 311, 314, 360 cover, 347, 348, 380 distance, 382, 385 metric, 304–308, 314, 316, 320, 321,

332, 334, 341, 347, 349, 374, 380–382, 393

Rolling, 312 Ruse’s invariant, 382, 389 S Second fundamental form, 316–319, 340 Sectional curvature, 322, 374, 380, 390 Semi-martingale, 282, 285, 289, 290, 296,

299, 311 Sphere, 285, 296, 312, 319, 326, 335,

338–340, 356, 380, 381, 390 Stable manifold, 334, 348, 349, 352, 354,

355 Standard Brownian motion with drift,

351 Stochastically complete, 322, 357, 360,

403 Stochastic development, 307–316 Stochastic mechanical flows, 334 Stochastic parallel translation, 311 Strictly conservative, 294 Strongly complete, 294, 295, 327 Structure equations, 329

Submanifold, 281, 283, 284, 286, 287, 304, 316, 317, 321, 324, 343, 352, 380

Supertrace, 407–408, 411 Symmetric space, 315, 335, 355 T Tangent

bundle, 283, 297, 298, 327, 356 space, 283, 284, 296, 297, 304, 309,

312, 319, 324, 341, 349 vector, 283, 304, 325, 353

Tensor bundle, 301, 356, 360, 367, 399 Torsion

form, 323–326 free, 299, 301, 303, 306, 321, 325, 328 tensor, 301, 325

Trivial bundle, 356, 360 Tubular neighbourhood, 284, 286, 290 V Vector field, 287, 288, 296, 300, 302, 303,

305–308, 310, 313, 316–321, 323–329, 334, 336, 362, 367, 381, 397, 400, 401

along a path, 305, 310, 327, 328, 381 Vertical tangent bundle, 297 Volume element, 331, 382 W Weitzenbock formula, 280, 363, 367–369,

398–402 W.K.B. approximation, 394