Divergence operators, transformations ofmeasure, and the interpolation method

Denis Bell

October 2, 2016

Transformations of measure

Given a Borel measure γ on a Banach space E and a measurabletransformation T : E 7→ E , define the induced measure νT by

γT (B) = γ(T−1(B)), B ∈ B(E )

We are interested in finding conditions on T such that γT γand computing the Radon-Nikodym derivative (RND) dγT

dγ whenthese conditions hold.

The Girsanov Theorem

Let w be a standard Wiener process in Rn, defined on the timeinterval [0,T ], and γ its law (the Wiener measure). Leth : [0,T ]× Ω 7→ Rn be a real-valued continuous bounded adaptedprocess. Then the process

wt = wt +

∫ t

0hsds, t ∈ [0,T ]

is a standard Wiener process with respect to the measure γ, where

d γ

dγ(w) = e−

∫ T0 hs ·dws− 1

2

∫ T0 |hs |

2ds

In the special case when h is a deterministic path, the Girsanovtheorem reduces to the Cameron-Martin theorem, which statesthat translating Wiener measure by a deterministic path h of finiteenergy, i.e. an absolutely continuous path such that∫ T

0|hs |2ds

produces an equivalent measure.

(The set H of such paths h is known as the Cameron-Martinspace.)

It is interesting to note that translation of Wiener measure by adeterministic path in Hc results in a singular measure.

Abstract Gaussian measures

Let H and E be, respectively, Hilbert and Banach spaces andi : H 7→ E be an abstract Wiener space in the sense of Gross withGaussian measure γ on E .

We identify E ∗ as a subspace of E under the embeddings

i∗ iE ∗ 7→ H∗ ∼= H 7→ E

There is a series of results concerning transformation of γ undermap T : E 7→ E , due to Kuo, Ramer, Kusuoka, Ustunel & Zakai...with the seminal result in this area due to Ramer.

Kuo’s theorem

Let T be a diffeomorrphism of E of the form

T = I + K

where I is the identity map on E and K is a C 1 map from E toE ∗. Then γT ∼ γ and

dγTdγ

(x) = | detDT (x)|−1e−(K(x),x)− 12|K(x)|2H

where (·, ·) denotes the pairing between E ∗ and E and det isdefined with respect to H.

Note the difference in hypotheses between this result and GT. Thelatter assumes the adapted condition, whereas in Kuo’s theorem(and like results), adaptedness is replaced by a C 1 assumption onK .

Divergence operators

Let γ denote a Borel measure on a Banach space E and U asubset of the set of vector fields on E .

DefinitionA linear operator L : U 7→ L2(γ) is said to be a divergence operator(DO) for γ if the following relation holds for all Z ∈ U and a denseclass of test functions φ on E∫

EDZφ(x)dγ =

∫Eφ(x)LZ (x)dγ.

Examples

Theorem(Gaveau-Trauber). The Ito integral

Lh =

∫ T

0hsdws

is a DO for the Wiener space. The domain of L is the set ofadapted process h of finite energy such that

E[ ∫ T

0|hs |2ds

]<∞.

Theorem(Gross). Let (H,E , γ) be an abstract Wiener space. Then theoperator

LZ (x) = (Z (x), x)− traceHDZ (x)

is a DO for γ, with domain the set of C 1 maps from E to E ∗.

In view of the Gaveau-Trauber theorem, this operator correspondsto the Ito integral in the classical adapted case. It can be identifiedwith the Skorohod integral. Nualart & Pardoux used this operatorto develop a theory of anticipating stochastic integration.

Relationship with Malliavin calculus

Suppose L is a DO for the space (E , γ) (e.g. a Gaussian space)and g : E 7→ K that is ”differentiable”. Assume that Z is a vectorfield on K for which there exists a lift R of Z to E , i.e.

Dg(x)R = Z .

Then∫KDZφ(x)dγg =

∫EDφ(g(x)Zdγ =

∫ED(φ g)(x)Rdγ

=

∫E

(φ g)LRdγ =

∫KφE [LR/g ]dγg .

i.e. DivγgZ = E [LR/g ].

The interpolation method

Let (L,U) denote a DO for γ on E and let T : E 7→ E be a map ofthe form T = IE + K , where K ∈ U. Define

Tt = I + tK , t ∈ [0, 1].

Suppose that Tt is invertible for all t. Note that γTt γ if and

only if there exists a family Xt of RND’sdγTtdγ .

Let φ be a test function on E . Then∫Eφ Tt(x)dγ =

∫EφXtdγ

Replacing φ by φ T−1t , we have

f (t) ≡∫Eφ T−1t (x)Xt(x)dγ =

∫Eφ(x)dγ.

Thus f is constant.

Computing f ′ by formally differentiating wrt in the first integral,we have

0 =

∫E

Dφ(T−1t (x))Xt(x) + φ T−1t (x)d/dtXt(x)

dγ.

We rewrite the first term in the integrand using the easily verifiedrelation

Dφ(T−1t (x))d/dtT−1t (x) = −D(φ T−1t )(x)K T−1t (x)

to get∫E

φT−1t (x)d/dtXt(x)−D(φT−1t )(x)KT−1t (x)Xt(x)

dγ = 0.

Assume the term (K T−1t )Xt ∈ U. We use the defining propertyof L to transform the second term in the integrand. This yields

∫Eφ T−1t (x)

d/dtXt(x)− L[(K T−1t )Xt ](x)

dγ = 0.

This will hold for all test functions φ if and only if Xt satisfies thedifferential equation

d/dtXt(x) = L[(K T−1t )Xt ](x).

We now make use of the following easily verified relation.

LemmaAssume h ∈ U ∩ L2(γ), ψ : E 7→ R ∈ C 1, ψh ∈ U. Then

L(ψh)(x) = ψ(x)Lh(x)− Dψ(x)h(x), a.s.(γ).

Using this above, we have

d/dtXt(x) = Xt(x)L[K T−1t ](x)− DXt(x))K Tt(x).

Denote Xt(x) by X (t, x), ∂X/∂t by X1 and ∂X/∂x by X2, andsubstitute x = Tt(y) to obtain

X1(t,Tt(y)) + X2(t,Tt(y)K (y) = X (t,Tt(y))L[K T−1t ](Tt)(y)).

Since K = d/dtTt , this is equivalent to

d/dtX (t,Tt(y)) = X (t,Tt(y))L[K T−1t ]Tt(y).

In view of the fact that X (0, ·) = 1, we have

X (t,Tt(y)) = exp

∫ t

0L[K T−1s ](Ts(y)ds.

Finally,

dγTdγ

= X (1, x) = exp

∫ 1

0L[K T−1s ](Ts T−1(x))ds.

Using this formula we can recover the RND’s in both the Girsanovtheorem and Kuo’s theorem, from the appropriate DO’s.

Note that, in principle, all the steps in the above argument arereversible. The idea is to start with a family of RND’s defined by

X (t,Tt(y)) = exp

∫ t

0L[K T−1s ](Ts(y)ds, t ∈ [0, 1]

and by reversing the steps, to obtain∫Eφ Tt(x)dγ =

∫EφXtdγ

thereby proving that γTt γ anddγTtdγ = Xt .

We used this this idea to obtain a generalization (to non-Gaussianmeasures) of the Cameron-Martin Theorem.

Further results in this direction were proved by Daletskii-Belopolskaya, Weizsacker, Smolyanov and others.

One problem is the invertibility condition on the familyTt , t ∈ [0, 1].

This holds if K ∈ L(E ) and is a strict contraction, i.e. there existsc ∈ (0, 1) with

||K (x)||E < c ||x ||E .

In this case Tt ∈ GL(E ) and

(I + tK )−1 =∞∑n=0

tnKn.

Change of measure under the flow of a vector field

Let E be a Banach space or a smooth manifold, equipped with afinite Borel measure γ and let Z be a C 1 vector field on E .

Consider the corresponding flow on E , i.e. the maps ρs : x 7→ xsdefined by

dxsds

= Z (xs),

x0 = x .

These maps are naturally diffeomorphisms of E (ρ−1s = ρ−s).

Say that Z is admissible if Z admits a divergence L. We say that γis quasi-invariant under (the flow of) Z if γs ≡ γρs ∼ γ for alls ∈ R.

Formally, quasi-invariance implies admissibility: suppose q-i holdsand denote dγs

dγ = Xs . Since ρ(0) = I and ρ′(0) = Z , we have

d

ds

/s=0

∫Eφdγs =

d

ds

/s=0

∫Eφ ρsdγ =

∫EDZφdγ.

Also

d

ds

/s=0

∫Eφdγs =

d

ds

/s=0

∫EφXsdγ =

∫EφdXs

ds

/s=0

dγ.

Hence Z is admissible and

DivZ =dXs

ds

/s=0

The converse is not true in general.

Example. Consider the case where E is Rn, dγ = Fdx , were F isC 1 with compact support, and Z = h, a constant vector in Rn.

Then Z is admissible with divergence −DhF/F . But not q-i underthe flow ρs(x) = x + sh.

Note, in this case LZ (x) blows up as x approaches the boundary ofthe support of F .

Using the interpolation method, we proved (CRAS, 2006):

TheoremSuppose Z is admissible with divergence LZ . Assume that thereexists B ⊂ E with γs(B) = 1, ∀s, such that the functions 7→ (LZ )(x−s) is continuous for x ∈ B. Then γ is quasi-invariantunder Z and

dγsdγ

(x) = exp

∫ s

0(LZ )(x−u)du.