Basic Properties of Brownian Motion

1 Stat205B: Probability Theory (Spring 2003)Lecture: 15 Basic Properties of Brownian MotionLecturer: James W. PitmanScribe: Rui this lecture, we discuss some Basic Properties of Brownian Motion , including various transformations, thetransition semigroup and its Motion lies in the intersection of several important classes of processes. It is a gaussian Markovprocess, it has continuous paths, it is a process with stationary independent increments (a L evy process),and it is a martingale. Several characterizations are known based on these consider also the following variation of Brownian Motion :Example a Brownian Motion (Bt, t 0) starting from 0.

2 LetXt=x+ t+ Bt, then (Xt, t 0)is a gaussian processes, all its FDDs (finite dimensional distributions) are multivariate normal. NotethatXis a Markov process, with stationary independent increments, withxthe initial state, the driftparameter, 2the variance parameter. These three parameters determine all the FDDs of (Xt, t 0), whichmay be called a Brownian Motion started atxwith drift parameter and variance parameter that the FDDs of a gaussian process (Xt, t I) are uniquely determined by the mean functiont E(Xt) and the covariance function (s, t) Cov(XsXt). Notice the covariance function must besymmetric and non-negative this idea, we have a second equivalent definition of Brownian Motion which is useful:Definition real valued process (Bt, t 0) is a Brownian Motion starting from 0 iff(a) (Bt) is a gaussian process;(b)EBt= 0 andEBsBt=s t, for alls, t 0;(c) With probability one,t Btis definition is often useful in checking that a process is a Brownian Motion , as in the transformationsdescribed by the following examples based on (Bt, t 0) a Brownian Motion starting from (scaling).

3 For eachs >0, (s 1/2 Bst, t 0) is a Brownian Motion starting from 0. This iseasy to verify if we use the definition above. Moreover, for eachs >0, (Bst, t 0)d= (s1/2Bt, t 0), allthe FDDs are the (shifting).For eachs >0, (Bs+t Bs, t 0) is a Brownian Motion starting from 0, andthis Brownian Motion is independent of (Bu,0 u s). This form of the Markov property of Brownianmotion ofBfollows easily from the stationary independent increments (time reversal).Consider (Bt,0 t 1), define (Xt,0 t 1) byXt=B1 t B1, then(Xt,0 t 1)d= (Bt,0 t 1).Example (inversion).The process (Xt, t 0) defined byX0= 0 andXt=tB(1/t) fort >0 is aBrownian Motion starting from 0.

4 To see this, notice (a) (Xt, t 0) is a gaussian process; (b)E(Xt) = 0,and ifs < tthenE(XsXt) =stE(B(1/s)B(1/t)) =s12 Basic Properties of Brownian Motion (c)Xclearly has paths that are continuous intprovidedt >0. To handlet= 0, we noteXhas the same FDDon a dense set as a Brownian Motion starting from 0, then recall in the previous work, the construction ofBrownian Motion gives us a unique extension of such a process, which is continuous att= 0. An alternativemethod is the following:Direct proof of strong law of large numbers,Bn/n 0 asn through the handle values between integers, use Kolmogorovs inequality,P( sup0<k 2m|B(n+k2 m) Bn|> n2/3) n 4/3E(Bn+1 Bn)2 Letm we getP( supu [n,n+1]|Bu Bn|> n2/3) n 4/3 Since nn 4/3< , the Borel-Cantelli lemma impliesBu/u 0 asu.

5 Takingu= 1/t, we haveXt 0 ast , letP0be the Wiener measure the distribution of (Bt, t 0) starting fromB0= 0, and letPxbethe distribution of (x+Bt, t 0). Consider the Brownian Motion as a time-homogenous Markov processwith transition kernelpt(x, dy) =1 2 te (y x)22tdyGenerally, given a group of probability kernels{pt, t 0}, we can define the correspondingtransitionoperatorsasPtf(x) := pt(x, dy)f(y) acting on bounded or non-negative measurable functionsf. Thereis an important relation between these two things:Theorem (semigroup property).The probability kernels{pt, t 0}, satisfy the Chapman-Kolmogrovrelation iff the corresponding transition operators{Pt}have the semigroup propertyPs+t=Ps Pts, t each bounded and measurablef,Pt+sf(x) = ps+t(x, dy)f(y)= ps(x, dz) pt(z, dy)f(y) Chapman-Kolmogrov relation= ps(x, dz)Pt(f(z)) = (Ps Pt)f(x)If{pt, t 0}is the family of transition kernels of a Markov process, the Markov property guaranteesthe Chapman-Kolmogrov relation, so the family of opertators{Pt}associated with a Markov process is asemigroup.

6 ThegeneratorQof the transition semigroup{Pt}is an operator defined asQf:= limt 0 Ptf ftfor suitablef, meaning that the limit exists in some Properties of Brownian Motion3 Now consider the semigroup of transition operators{Pt}and its generator for Browning Motion . By defini-tion,Ptf(x) = Rpt(x, y)f(y)dy= 1 2 te (y x)22tf(y)dy= 1 2 e z22f(x+ tz)dzAs for the generator, forfwith two continuous derivativesf andf such thatf is bounded,Qf(x) = limt 0 Ptf(x) f(x)t= limt 0 1 2 e z22f(x+ tz) f(x)tdz= limt 0 1 2 e z22f (x) tz+f (x+ tz)tz2/2tdz= limt 0 1 2 e z22f (x+ tz)z22dz=12f (x)where [0,1] is function ofxand tz, so ast 0 there is the convergence tz 0, hencef (x+ tz) f (x) by continuity off , and the last step is justified by the dominated convergence theorem, using theassumption thatf is Durrett.

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