Complex Analysis and Brownian Motion › ~morrow › 336_13 › ... · Complex Analysis and...

Complex Analysis and Brownian Motion

Yuxuan Zhang

June 5, 2013

Abstract

This paper discusses some basic ideas of Brownian motion. Beginningfrom measure theory, this paper makes a brief introduction to stochasticprocess, stochastic calculus and Markov property, recurrence as well asmartingale related to Brownian motion. Later, it shows an applicationof Brownian motion which applies Brownian motion to prove Liouville’stheorem in complex analysis.

Contents

1 Introduction 2

2 Brownian Motion 32.1 Sigma Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Some related properties for Brownian motion . . . . . . . . . . . 6

2.4.1 Markov property . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Stochastic Calculus 93.1 Introduction to Stochastic Calculus . . . . . . . . . . . . . . . . . 93.2 Ito Process and Ito’s Formula . . . . . . . . . . . . . . . . . . . . 10

4 The Application of Brownian Motion 124.1 Proof of Liouville’s theorem from Brownian motion . . . . . . . . 12

1

Complex Analysis and Brownian Motion 2

1 Introduction

The first observation to Brownian Motion was in 1827 by British botanist,Robert Brown. When studying pollen grains under the microscope, he sur-prisingly found that the pollen grains are not static but instead, move in someirregular way. However, at that time, Brown cannot find out the mechanismswhich caused this strange motion. The puzzle existed for about 100 years un-til Albert Einstein, who published a related paper in 1905, suggested that themotion which Brown observed was caused by the impact from the moving wa-ter molecules. Molecules constantly bombard the pollen grain and due to theimbalance force from each direction, the pollen grain will move in a constantlychanging path. Brownian motion is a great example which we can directlyobserve the consequence of the moving of unobservable particles and this obser-vation provides a solid confirmation for the existence to the molecules and atoms.

Fig 1 A three dimensional Brownian Motion for times 0 ≤ t ≤ 2 [4].

Scientists have studied Brownian Motion for a long time, and mathematiciansalso, from their perspective, provide their explanation and prediction result toBrownian Motion. Brownian Motion now is not a mysterious observation toconfuse people any more, but instead, we apply what we get from the study toBrownian Motion to help us determine more and more complex phenomenonin this world. For example, the use of Brownian Motion to predict the Stockmarket [5] and the application in the prediction of heat flow [1]. In this paper,we will discuss the study of Brownian Motion structured in math related tocomplex analysis and later, we will consider some examples related to BrownianMotion.


2 Brownian Motion

In this section, we’ll cover up some definition and basic properties for BrownianMotion.

2.1 Sigma Algebra

As the steal theoretical foundation of the modern probability, the measure the-ory provides us a pure mathematical perspective of probability knowing fromthe classical, frequency or subjective interpretations to probability from philos-ophy. We’ll here only discuss some basic theorems building up the whole system.

Definition 2.1.1 (σ − algebra) A collection F of subsets of a set X iscalled a σ − algebra if

• 1. ∅ ∈ F ;

• 2. if A ∈ B, then A′ ∈ F ;

• 3. if A1, A2, ... , An,... ∈ B, then⋃∞i=1Ai ∈ F .

The pair (X,F) is a field of sets, called a measurable space.

Next, we will give a formal definition of the probability space as well as theprobability measure.

Definition 2.1.2 (Probability Measure) A probability measure on a givenprobability space is function υ satisfying the following conditions:

• 1. υ is a function map event space to unit interval [0, 1], or υ : Ω 7→ [0, 1];

• 2. υ(⋃i∈I Ei) =

∑i∈I υ(Ei)

Definition 2.1.3 (Probability Space) A probability space is a triple (Ω, F ,P ) consisting of:

• the sample space Ω as an arbitrary non-empty set;

• the σ-algebra F ⊆ 2Ω ;

• the probability measure P : F 7→ [0, 1] .

And the following is the formal definition of filtration, which will be usedlater when defined stopping time and martingale.

Definition 2.1.4 (Filtration) A filtration on (Ω,F , P ) is a collection mea-surable sets Ft : t ≥ 0 which satisfies Fs ⊂ Ft ⊂ F if s < t.


2.2 Stochastic Process

Also called as random process, the stochastic process is often used to show theevolution of some random value based on time. Not like how deterministic processworks, which predicts the process can only develop in one way, stochastic pro-cess allows some indetermination.

Definition 2.2.1 (Stochastic Process) Given a probability space (Ω,F , P )and a measurable space (S,Σ), an S-valued stochastic process is a collection ofS-valued random variables on Ω, indexed by a totally ordered time set T. Thatis, a stochastic process B is a collection

Bt : t ∈ T

where each Bi is an S-valued random variable on Ω. The space S is then calledthe state space of the process.

In the study of stochastic process, there is a important concept called stoppingtime. As a specific type of ”random time”, stopping time is a random variablerelated to time with respect to the event space Ω. The following is the formaldefinition of stopping time based on filtration.

Definition 2.2.2 (Stopping time) A random variable T : Ω 7→ [0,∞] de-fined on a filtered probability space is called a stopping time with respect to thefiltration F if the set x ∈ Ω : T (x) ≤ t ∈ Ft for all t.

One of the example of stopping time is the first occasion of the expected event.From stopping time, we can decide whether T ≤ t simply by knowing the statesof the stochastic process until time t [6].

2.3 Brownian Motion

Now, based on our theories above, we’ll be able to give the formal definition ofBrownian motion based on measure theory.

Definition 2.3.1 (d − dimensional Brownian Motion) A d-dimensionalBrownian motion is a stochastic process Bt : Ω 7→ R from the probability space(Ω,F , P ) to Rd such that the following properties hold:

• 1. (Independent Increments) For any finite sequence of times t0 < t1 <... < tn, the distributions Bti+1 −Bti for i = 1, ..., n are independent,

• 2. For all ω ∈ Ω, the parametrization function t 7→ Bt(ω) is continuous,

• 3. (Stationary) For any pair s, t ≥ 0, let Bs+t −Bs ∈ A,

P (Bs+t −Bs) =

∫A

1

(2πt)d/2e−|x|

2/2tdx.


Standard Brownian motion is the Brownian motion where B0(ω) = 0.

For a better understanding to Brownian motion from intuition, let’s considerthe case where Brownian motion in the one dimension.

Definition 2.3.2 (Standard Brownian Motion in 1 dimension) A stan-dard Brownian motion in one dimension (Bt)t≥0 is a real-valued stochasticprocess defined on a probability space (Ω,F , P ) which satisfies the followingconditions:

• 1. B0 = 0,

• 2. For any finite sequence of times t0 < t1 < ... < tn, the distributionsBti+1 −Bti for i = 1, ..., n are independent,

• 3. For all ω ∈ Ω, t 7→ Bt(ω) is continuous,

• 4. For all s, t ≥ 0, Bs+t −Bs is independent of (Bu)0≤u≤s and has distri-bution N (0, t). Where N (µ, σ2) is the normal distribution

N (µ, σ2) =1

σ 2√

2πe−

(x−µ)2

2σ2

with mean µ and stand deviation σ.

The definition above is directly entailed from Definition 2.3.1. In this case, wecan more clearly get the intuition of Brownian motion. Especially the stationaryproperty, which, as a normal distribution, is much more messy in the higher di-mension case. Actually, from [8], we can treat d-dimensional Brownian motionas the combination of the easy case in one dimension as what we’ll show in thefollowing.

Definition 2.3.3 A Brownian motion in Rd is a d-dimensional vector whosecomponents are independent scalar Brownian motions.

It’s obvious that 2.3.1.1 and 2.3.1.2 are directly entailed by the similar state-ment in 2.3.2. The following proof will be focused on the transformation of thethird criteria, which constructs the d-dimensional normal distribution from dindependent normal distribution in 1 dimension.

Denote the ith component of x by xi, the joint probability density functionof d-dimensional normal distribution (X ∼ N (µ, σ2)) can be described in thefollowing way:

fX(x) =

∫X

1

(2πt)d/2e−|x|

2/2tdx

=

∫X

(1

(2πt)1/2)(

1

(2πt)1/2)...(

1

(2πt)1/2)︸︷︷︸

dtimes

e−x21/2t−x

22/2t−...−x

2d/2tdx1dx2...dxd


=

∫X1

1

(2πt)1/2e−x

21/2tdx1

∫X2

1

(2πt)1/2e−x

22/2tdx2...

∫Xd

1

(2πt)1/2e−x

2d/2tdxd

= f(x1)f(x2)...f(xn)

where f(xi) is the probability density function of a normal distribution and eachof f(xi) is independent to others.

2.4 Some related properties for Brownian motion

Now, we have a very basic idea about Brownian motion. In this section, we’llexplore some more interesting properties of Brownian motion for the later dis-cussion between Brownian motion and complex analysis.

2.4.1 Markov property

In probability theory, the Markov property refers to the memoryless property ofa stochastic process [10]. In other word, if a stochastic process has this property,the former process will not influence the later process. From the independent in-crements definition of Brownian motion, Brownian motion directly has Markovproperty from its definition.

Theorem 2.4.1 (Markov property) Let Bt : t ≥ 0 is a Brownian mo-tion started at x ∈ Rd. Fix t > 0, then the process Bt+s − Bt : s ≥ 0 is aBrownian motion starting at origin and independent of Bt : 0 ≤ t ≤ s. In otherword, Brownian motion satisfies Markov property.

Proof. This is directly entailed from the Definition of Brownian motion. Fromthe definition of Brownian motion (Definition 2.3.1), we know that Brownianmotion satisfies the independent increments property. That is, for any finitesequence of times t0 < t1 < ... < tn, the distributions Bti+1

−Bti for i = 1, ..., nare independent. Since the process

Bt+s −Bt =

s∑j=1

Bt+j −Bt+j−1 (s > 0),

where each term Bt+j − Bt+j−1 is independent, the given process is hence in-dependent.

2.4.2 Recurrence

In this section, we’ll introduce the recurrence property of Brownian motion.First is the definition of recurrence.

Definition 2.4.2 (Recurrence and Transience) Brownian motion Bt : t ≥ 0is:

• 1. transient if limt→inf |Bt| =∞


• 2. point recurrent if for every x ∈ Rd, there is an increasing sequence tnsuch that Btn = x for all n ∈ N

• 3. neighbourhood recurrent if for every x ∈ Rd and ε > 0, there exists anincreasing sequence tn such that Btn ∈ Bε(x) for all n ∈ N .

From the definition above, recurrence somehow gives us a criteria to definehow Brownian motion moves in the Rd space. It’s easy to figure out 1 dimen-sional Brownian motion has point recurrent from intuition. And according tothe naive intuition one may infer that in higher dimension the Brownian motionmay also be point recurrent as what is in the 1 dimension. But in fact, in higherdimension (especially when d ≥ 3), Brownian motion does not follow the samerule any more.

Theorem 2.4.3 Brownian motion is:

• 1. point recurrent in dimension d = 1,

• 2. neighbourhood recurrent, but not point recurrent, in dimension d = 2(also called as planar Brownian motion),

• 3. transient in dimension when d ≥ 3.

For our interest, we will only show the proof of the recurrence in the planarBrownian motion. 1

Proof. Let d = 2, 0 < r < R <∞ and U = x ∈ Rd : r < |x| < R.Meanwhile, let F (x) = P(|BtU | = R)|(B0 = x) be the probability that the

Brownian motion starting at x reaches BR before reaching 0. Then F satisfiesF (r) = 0, F (R) = 1 and ∇2F (x) = 0 for r < x < R. 2

By the symmetry, it’s easy to determine that ∃φ(|x|) = F (x) such thatφ(r) = 0, φ(R) = 1. From ∇2F (x) = 0,

∇2F (x) = ∇2F (x1, x2) =

d∑j=1

∂jjφ(√x2

1 + x22) = φ′′(|x|) +

d− 1

|x|φ′(|x|).

Therefore, we need the solution to this differential equation:

sφ′′ + φ′(s) = 0.

As a first-order linear differential equation in the terms of φ′, we can get thegeneral solution in the following form,

φ(s) = c1 log s+ c2,

1Some more proof related to other dimension can be found in [11].2∇2F = 0 is a conclusion from the solution to the DirichLet problem. Details can be found

at [6]


and when we plug in the boundary conditions φ(r) = 0, φ(R) = 1, we’ll get thesolution:

φ(|x|) =log |x| − log r

logR− log r,

so, we can get the following formula related to the Brownian motion on theannulus U = x ∈ Rd : r < |x| < R:

F (x) = P(|BtU | = R)|(B0 = x) =log |x| − log r

logR− log r.

Let r < |x| be fixed and let R → ∞, the probability that we’ll reach the Br isgiven by:

limR→∞

P(|BtU | = r)|(B0 = x) = limR→∞

log |x| − log r

logR− log r= 1.

This is to say, for every fixed r > 0, the Brownian motion will keep reachingthe Br as t→∞. However, since the probability of reaching 0 before reachingdistance R will be given by:

P(reach 0 before distance R) = limr→0

P(|BtU | = r)|(B0 = x)

= limr→0

log |x| − log r

logR− log r= 0,

which will never reach the origin. So according to the definition of recurrence,planar Brownian motion is only neighbourhood recurrent but not point recur-rent.

From the definition of the neighbourhood recurrence, we can directly detailthe following corollary:

Corollary 2.4.4 Neighbourhood recurrence implies the path of planar Brow-nian motion is dense in the plane.

2.4.3 Martingale

In probability, there is a kind of model of fair game in which the past eventshave nothing to do with predicting the future. We call this kind of game asmartingale. The formal definition of martingale is as following.

Definition 2.4.6 (Martingale) In general, a stochastic process B : T×Ω 7→S is a martingale with respect to a given filtration F∗ and probability measureP if:

• 1. F∗ is a filtration of the probability space (Ω,F , P ).

• 2. For each t in T , all each random variable Bt is measurable with respectto Ft. We call Bt is adapted to Ft.


• 3. E[Bt] is finite. Where E(X) is the expectation to the random variableX.

• 4. for any s, t ∈ T and 0 ≤ s ≤ t, E[Bt|Fs] = Bs.

There is also a concept called local martingale. We’ll need to use this later toprove the Levy’s characterization. The following is the definition of local martingale,

Definition 2.4.7 (Local Martingale) A local martingale refers to an adaptedstochastic process X(t) : 0 ≤ t ≤ T which contains a sequence of stopping timesTn such that X(mint, Tn) : t ≥ 0 is a martingale for every n.

With the definition of martingale, we can know expressing an important theo-rem we will use later.

Theorem 2.4.8 (Optional Stopping Theorem) Suppose Xt is a continuousmartingale and 0 ≤ S ≤ T are stopping times. If the process X(mint, T) isdominated by an integrable random variable Y , then E[XT |FS ] = XS .

3 Stochastic Calculus

The stochastic calculus is a mechanism used for operating on stochastic process.This kind of calculus is used to model systems which behaves randomly. In thissection, we will introduce some basic ideas about stochastic calculus as well asan important property of this integral named Ito’s formula.

3.1 Introduction to Stochastic Calculus

In the past integration we have ever met (Riemann Integration, Contour Inte-gration), we always discussed the mathematical process in a determined case, inwhich we need the function to be sufficiently nice. However, in Brownian mo-tion, these integration ways based on the nice function will not work so well dueto the extremely irregularity of Brownian motion itself. Instead of using meth-ods we use in the determined case, we will consider a new kind of integrationbased on the following differential equation:

dB

dt= v(t,Xt) + w(t,Xt)Wt,

where we bring a new variable Wt representing the ”randomness” of this func-tion. We define the Wt to have the following properties:

• 1. if t1 6= t2, then Wt1 and Wt2 are independent,

• 2. the set Wt, t ≥ 0 is stationary.

• 3. E[Wt] = 0 for all t.


For solving this differential equation, we will build up a new kind of integral,that is the stochastic integral. Let’s consider the difference based on a smalltime interval ∆t in the discrete case and then take limit to this interval get theintegrand just like what we did in Riemann integration before:

Xk+1 −Xk = Xtk+1 −Xtk = v(tk, Xk)∆tk + w(tk, Xk)Wk∆tk.

Now we’ll have new problem. It’s easy to understand what v(tk, Xk)∆tk is, butwhat does the second term w(tk, Xk)Wk∆tk represent? Consider our definitionto Wt, let Wk∆tk = Wktk+1−Wktk = Vk+1−Vk for some new stochastic processVt, then we can interpret Vt as a stochastic process with stationary independentincrement with mean 0. Sounds familiar? This is exactly our definition toBrownian motion (Definition 2.3.1). That is to say: Vk = Btk . Thus sum up allthe discrete part we will get:

Xk = X0 +

k−1∑j=0

v(tj , Xj)∆tJ +

k−1∑j=0

w(tj , Xj)∆Bj .

Convert it from discrete case to continuous case, we will get the following for-mula:

Xt = X0 +

∫ t

0

u(s,Xs)ds+

∫ t

0

w(s,Xs)dBs,

which left us is a final problem, how can we define dBs ? Actually, from thedefinition of Brownian motion, we can give the following definition of an inte-gration over Bs,

Definition 3.1.1 (Stochastic Integral with respect to Brownian Motion)The stochastic integral with respect to Brownian motion (Bt)t∈R+ of any mea-surable function f ∈ L2(R+) is defined by∫ ∞

0

f(t)dBt ∼ N (0,

∫ ∞0

|f(t)|2dt),

where N (µ, σ2) is the normal distribution with the average at µ, standard de-viation as σ.

This is easy to understand from the definition of Brownian motion. The in-dependent increments property ensures that every dBt is an independent pro-cess and furthermore, from the stationary property, each of these independentprocesses have the normal distribution. Recall that, if X1, X2, ..., Xn are in-dependent random variables (specially, Gaussian random variable here) withN(m1, σ

21), N(m2, σ

22), ...,N(mn, σ

2n), then the sum

∑ni=1Xi is a random vari-

able with N(∑ni=1mi,

∑ni=1 σ

2i ).

3.2 Ito Process and Ito’s Formula

One of the most important property of stochastic calculus is Ito’s formula. Ito’sformula is kind of like chain rule in the calculus we are familiar with. Before we


give the statement of Ito’s formula, we need the definition of Ito process:

Definition 3.1.2 (Ito Process) An Ito process itself is a stochastic pro-cess It on (Ω,F , P ) of the form:

It = I0 +

∫ t

0

u(s, w)ds+

∫ t

0

v(s, w)dBs,

where v ∈ V , P (∫ t

0v2(s, w)ds < ∞,∀t ≥ 0) = 1, u is Ft - adapted and

P (∫ t

0|u(s, w)|ds <∞,∀t ≥ 0) = 1). The equation above can be written as:

dIt = udt+ vdBt.

The first term is called drift and the second term is called volatility.With the definition of Ito process, the Ito’s formula itself can be described

as below:

Theorem 3.1.3 (Ito′s Formula) Let It be an Ito progress given in 3.1.1.Let g(t, I) ∈ C2([0,∞)× R). Then Yt = g(t, It) is again an Ito process and:

dYt = g′t(t, It)dt+ g′i(t, It)dIt +1

2g′′ii(t, It)(dIt)

2,

where(dIt)

2 = (udt+ vdBt)2 = v2dt.

Substitute dIt we can get:

dYt = (g′t + g′iu+1

2g′′iiv

2)dt+ g′xvdBt.

With the first term as the new drift and the second term as the new volatilityin the formula above. 3

From Ito’s formula, we can get the following theorem which will be used later,

Theorem 3.1.4 Let D ⊂ Rd be a connected open set and f : D → Rbe harmonic on D. Let Bt with 0 ≤ t ≤ T be Brownian motion starts inside Dand stops at T , then process f(Bt) : 0 ≤ t ≤ T is a local martingale.

Proof. Let f be a C2 function. Fix any compact K ⊂ U . Meanwhile, letg : Rm → [0, 1] be a smooth function inside U such that g = 1 on K. Definef∗ = fg and apply Ito’s formula to mins, T for all s < T where T is the exittime of K. Let Bs be an Brownian motion and Xs be a stochastic process,according to Ito’s formula,

f(B(mint, Tn), X(mint, Tn))− f(B0, X0) =

∫ mint,Tn

0

∇xf(Bs, Xs)

3The detailed proof of this theorem can be found at [14].


+

∫ mint,Tn

0

f(Bs, Xs)dXs +1

2

∫ mint,Tn

0

4xf(Bs, Xs)ds.

Now suppose that Kn be a set such that K1 ⊂ K2 ⊂ ... ⊂ Kn such that∪n∈NKn = D. Let Tn be the exit time of Kn, Xs = 0, ∇yf(Bs) = 0. Thensince f is harmonic, 4xf = 0, and therefore the equation above now turns to,

f(Bmint,Tn) = f(B0) +

∫ mint,Tn

0

∇f(Bs)dBs.

Therefore, we have found an increasing sequence of stopping time converging tothe final stopping time T . So f(B(mint, Tn)) is a local martingale.

4 The Application of Brownian Motion

With all the theorems constructed before, we can use them to explore somethinginteresting. In this section, by applying what we get before, we will see a coolapplication from Brownian Motion to prove the Liouville’s theorem in complexanalysis.

4.1 Proof of Liouville’s theorem from Brownian motion

The following is Liouville’s Theorem:

Theorem 4.1.1 (Liouville′s Theorem) Suppose f is a complex valued func-tion that is entire and bounded, then it is constant.

For proving this theorem, we still need the help of two more theorems whichcan be deduced from Ito’s formula. One of them are Levy’s theorem and theother is Dubins and Schwarz Corollary to Levy’s theorem.

Theorem 4.1.2 (Levy′s Theorem) Suppose that both M and (M2t − t)t≥0

are local martingales. Assume M0 = 0. Then M is a Brownian motion withrespect to (Ft).

Proof. Let f(x) = eivx where v ∈ R. Since f ∈ C2(R), then according to Ito’sformula,

f(Mt) = f(0) +

∫ t

0

f ′(Ms)dMs +1

2

∫ t

0

f ′′(Ms)ds,

where Mft :=

∫ t0f ′(Ms)dMs is a local martingale. Further more, if f ′ and f ′′

are bounded, then Mft is a martingale. So when taking expectations of the

formula above, we will get,

E[f(Mt)] = f(0) +1

2

∫ t

0

E[f ′′(Ms)]ds.


Let g(t) = E[f(Mt)], then by substituting f(Mt) in the integration,

g(t) = 1− v2

2

∫ t

0

g(s)ds.

From the formula above, we can get that g(t) is the solution to a differentialequation which satisfies the following initial conditions,

g′(t) = −v2

2g(t) and g(0) = 1.

As a first order differential equation, the unique solution to g is

g(t) = e−tv2

2 .

So,

E[eivMt ] = e−tv2

2 ,

which shows that Mt is normally distributed with mean 0 and variance t12 .

Now let s > 0 and A ∈ Fs with P (A) > 0. Let P ∗(B) = P (B|A), F∗t = Ft+sand M∗t = Mt+s −Ms for t ≥ 0. Then with respect to F∗t over probabilityspace (Ω,F , P ∗), (M∗t )t≥0 is continuous local martingale with M∗0 = 0 suchthat [M∗t ]2 − t is also a local martingale. Then from what we already prove,

E[eivM∗t ] = e−

tv2

2 .

Substitute M∗t = Mt+s −Mt and let A varies in Fs,

E[eiv(Mt+s−Mt)|Fs] = e−tv2

2 ,

which exactly shows that Mt+s − Ms is independent of Fs and is normallydistributed. Hence Mt+s −Ms is Brownian motion.

The corollary from Dublins and Schwarz is described as below.

Corollary 4.1.4 (Dubins and Schwarz) Let M be a continuous local mar-tingale null at 0 such that [M ]t is increasing as t → ∞. For t ≥ 0, we definestopping times τt := inf u : [M ]u > t and a shifted filtration G(t) = F(τt). ThenX(t) = M(τt) is standard Brownian motion.

Proof. For any rational q, let

Sq = inft > q|[M ]t > [M ]q.

Then we can get the following expectation,

E[M(Sq)2 − [M ](Sq)|Fq] = M(q)2 − [M ]q.

According to the definition of infimum, [M ](Sq)2 = [M ]q, so the expectation is

0 and M is constant on [q, Sq). Since Xt = M(τt), X is continuous. Then if we


can prove that X and X2 − t are local martingales, then according to Levy’sTheorem, X is Brownian motion. Firstly, let’s define

T (n) = inft||Mt| > n and U(n) = [M ](T (n)),

Then for each t, τmint,U(n) = minT (n), τt. So if we let 1 be the indicatorfunction, then

X(mint, U(n)) = M(τt)1τt≤T (n).

Apply optional stopping theorem to M1t≤T (n), for s < t, we will get

E[X(mint, U(n))|Gs] = E[M(τt)1τt≤T (n)|Gs]

= M(τt)1τt≤T (n) = X(mins, U(n)).

So, according to the definition of local martingale, Xt is local martingale. Andby the same process, we can prove that Xt − t is also local martingale. Soaccording to Levy’s Theorem, X is Brownian motion.

Now, we have enough tools to prove the Liouville’s theorem.

Proof. Suppose f is an entire function but not constant. Then according toTheorem 3.1.4, when given a Brownian motion Bt, f(Bt) will be a local mar-tingale. Then by Corollary 4.1.4, f(Bt) is also a Brownian motion. Since we’vealready proved that the planar Brownian motion is neighbourhood recurrent inTheorem 2.4.3, which indicates that f is dense in the whole complex plane andhence f is not bounded. This is contradiction to what we assume. So the entirefunction must be constant.


References

[1] Gregory Lawler. "Random Walk and the Heat Equation."

Department of Mathematics, University of Chicago, n.d. Web.

[2] Stochastic process,

http://en.wikipedia.org/wiki/Stochastic process, web.

[3] Sigma algebra,

http://en.wikipedia.org/wiki/Measurable space, web.

[4] Brownian motion,

http://en.wikipedia.org/wiki/Brownian motion, web.

[5] Simulation of Stock Price in Excel,

http://www.youtube.com/watch?v=oOnD S2jq4U, web.

[6] Brownian Motion And Liouville’s Theorem, Chen Hhi George Teo,

http://math.uchicago.edu/∼may/REU2012/REUPapers/Teo.pdf, web.

[7] Introduction to probability and measure, Parthasarathy, K.R.,

2005, Hindustan Book Agency.

[8] Brownian motion, complex analysis, and the dimension of the

Brownian frontier, Sam Watson, 2010,

http://math.mit.edu/∼sswatson/ pdfs/partiiiessay.pdf, web.

[9] Multivariate normal distribution,

http://www.statlect.com/mcdnrm1.htm, web.

[10] Markov Property,

http://en.wikipedia.org/wiki/Markov property, web.

[11] Stochastic Calculus, Alexey Kuptsov, New York University,

https://files.nyu.edu/ak1103/public/Teaching/Summer2011/Lecture8 Summer2011.pdf,

web.

[12] Martingale (Probability theory)

http://en.wikipedia.org/wiki/Martingale (probability theory),web.

[13] Levy’s Theorem, UCSD,

http://math.ucsd.edu/∼pfitz/downloads/courses/spring05/math280c/levy.pdf,web.

[14] Brownian Motion and Stochastic Calculus, NTU,

http://www.ntu.edu.sg/home/nprivault/MA5182/chapter4.pdf,web.

Complex Analysis and Brownian Motion › ~morrow › 336_13 › ... · Complex Analysis and...

Documents

Transcript of Complex Analysis and Brownian Motion › ~morrow › 336_13 › ... · Complex Analysis and...