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Brownian motion by krzysztof burdzy(university of washington)
 

Brownian motion by krzysztof burdzy(university of washington)

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    Brownian motion by krzysztof burdzy(university of washington) Brownian motion by krzysztof burdzy(university of washington) Presentation Transcript

    • BROWNIAN MOTIONA tutorialKrzysztof BurdzyUniversity of Washington
    • A paradox∞<′′→∈|)(|sup,]1,0[:]1,0[tfRft(*)))((21exp)()10,)()((102′−≈<<+<<−∫ dttfcttfBtfP tεεε
    • (*) is maximized by f(t) = 0, t>0Microsoft stock- the last 5 yearsThe most likely (?!?) shape of aBrownian path:
    • Definition of Brownian motionBrownian motion is the unique processwith the following properties:(i) No memory(ii) Invariance(iii) Continuity(iv) tBVarBEB tt === )(,0)(,00
    • Memoryless process0t2t1t 3t,,, 231201 tttttt BBBBBB −−−are independent
    • InvarianceThe distribution ofdepends only on t.sst BB −+
    • Path regularity(i) is continuous a.s.(ii) is nowhere differentiable a.s.tBt →tBt →
    • Why Brownian motion?Brownian motion belongs to several familiesof well understood stochastic processes:(i) Markov processes(ii) Martingales(iii) Gaussian processes(iv) Levy processes
    • Markov processes}0,|,{}|,{ suBstBBstB utst ≤≤≥=≥ LLThe theory of Markov processes usestools from several branches of analysis:(i) Functional analysis (transition semigroups)(ii) Potential theory (harmonic, Green functions)(iii) Spectral theory (eigenfunction expansion)(iv) PDE’s (heat equation)
    • Martingales are the only family of processesfor which the theory of stochastic integrals isfully developed, successful and satisfactory.Martingalessst BBBEts =⇒< )|(∫tss dBX0
    • Gaussian processesnttt BBB ,,, 21 is multidimensionalnormal (Gaussian)(i) Excellent bounds for tails(ii) Second moment calculations(iii) Extensions to unordered parameter(s)
    • The Ito formula∑∫ =+∞→−=ntknknknkntss BBXdBX0//)1(/0)(lim∫ ∫ ′′+′+=t tssst dsBfdBBfBfBf0 00 )(21)()()(
    • Random walkIndependent steps, P(up)=P(down){ } { }0,0,/ ≥ →≥ ∞→tBtWa taat(in distribution)tWt
    • ScalingCentral Limit Theorem (CLT),parabolic PDE’s}10,{}10,{ / ≤≤=≤≤ tBatB atDt
    • The effect is the same as replacingwithMultiply the probability of each Brownian pathbyCameron-Martin-Girsanov formula}10,{ ≤≤ tBt′−′∫ ∫10102))((21)(exp dssfdBsf s}10,{ ≤≤ tBt }10),({ ≤≤+ ttfBt
    • Invariance (2)}10,{}10,{ 11 ≤≤−=≤≤ − tBBtB tDtTime reversal0 1
    • Brownian motion and the heat equation),( txu – temperature at location x at time t),(21),( txutxutx∆=∂∂Heat equation:dxxudx )0,()( =µ)(),( dyBPdytyu t ∈= µForwardrepresentation)0,(),( yBEutyu t +=Backward representation(Feynman-Kac formula)0 tµy
    • Multidimensional Brownian motion,,, 321ttt BBB - independent 1-dimensionalBrownian motions),,,( 21 dttt BBB  - d-dimensional Brownianmotion
    • Feynman-Kac formula (2))(xf−= ∫ττ0)(exp)()( dsBVBfExu sxτB x0)()()(21=−∆ xuxVxu
    • Invariance (3)The d-dimensional Brownian motion is invariantunder isometries of the d-dimensional space.It also inherits invariance properties of the1-dimensional Brownian motion.)2/)(exp(21)2/exp(21)2/exp(2122212221xxxx+−=−−πππ
    • Conformal invariancef}0),()({ 0 ≥− tBfBf tanalytictB )( tBfhas the same distribution as∫ ′=≥tstc dsBftctB02)( |)(|)(},0,{
    • If thenThe Ito formulaDisappearing terms (1)∫ ∫∆+∇+=t tssst dsBfdBBfBfBf0 00 )(21)()()(∫∇+=tsst dBBfBfBf00 )()()(0≡∆f
    • Brownian martingalesTheorem (Martingale representation theorem).{Brownian martingales} = {stochastic integrals}},{,)|(0tsBFMMMMEdBXMsBttssttsst≤=∈== ∫σ
    • The Ito formulaDisappearing terms (2)∫∫∫∂∂+∂∂+∂∂=−tststsstdsBsfxdsBsfsdBBsfxBtfBtf022000),(21),(),(),(),(∫∫ ∂∂+∂∂=−tststdsBsfxEdsBsfsEBtEfBtEf02200),(21),(),(),(
    • Mild modifications of BMMild = the new process correspondsto the Laplacian(i) Killing – Dirichlet problem(ii) Reflection – Neumann problem(iii) Absorption – Robin problem
    • Related models – diffusionsdtXdBXdX tttt )()( µσ +=(i) Markov property – yes(ii) Martingale – only if(iii) Gaussian – no, but Gaussian tails0≡µ
    • Related models – stable processes2/1)(dtdB =α/1)(dtdX =Brownian motion –Stable processes –(i) Markov property – yes(ii) Martingale – yes and no(iii) Gaussian – noPrice to pay: jumps, heavy tails, 20 ≤<α220 ≤<
    • Related models – fractional BMα/1)(dtdX =(i) Markov property – no(ii) Martingale – no(iii) Gaussian – yes(iv) Continuous∞<<α1∞<< 21
    • Related models – super BMSuper Brownian motion is related to2uu =∆and to a stochastic PDE.
    • Related models – SLESchramm-Loewner Evolution is a modelfor non-crossing conformally invariant2-dimensional paths.
    • (i) is continuous a.s.(ii) is nowhere differentiable a.s.(iii) is Holder(iv) Local Law if Iterated LogarithmPath propertiestBt →tBt →tBt → )2/1( ε−1|log|log2suplim0=↓ ttBtt
    • Exceptional points1|log|log2suplim0=↓ ttBtt1|)log(|log)(2suplim =−−−↓ ststBB ststFor any fixed s>0, a.s.,),0()(2suplim ∞∈−−↓ stBB ststThere exist s>0, a.s., such that
    • Cut pointsFor any fixed t>0, a.s., the 2-dimensionalBrownian path contains a closed looparound in every intervaltB ),( ε+ttAlmost surely, there existsuch that)1,0(∈t∅=∩ ])1,(()),0([ tBtB
    • Intersection properties1))((.,.)4(2))((.,.2))((.,.)3())((.,...)2(.,.)1(14131312≤∈∀=≤∈∀=∈∃=∞=∈∃≠≠∀∀==≠∃∀=−−−−xBCardRxsadxBCardRxsaxBCardRxsadxBCardRxsaBBtssatdBBtstsadtsts
    • Intersections of random sets∅≠∩>+BAdBA)dim()dim(The dimension of Brownian trace is 2in every dimension.
    • Invariance principle(i) Random walk converges to Brownianmotion (Donsker (1951))(ii) Reflected random walk convergesto reflected Brownian motion(Stroock and Varadhan (1971) - domains,B and Chen (2007) – uniform domains, not alldomains)(iii) Self-avoiding random walk in 2 dimensionsconverges to SLE (200?)(open problem)2C
    • Local timeststBtBMdsL s≤<<−→== ∫sup21lim }{0t01 εεε ε
    • Local time (2)}10,|{|}10,{}10,{}10,{≤≤=≤≤−≤≤=≤≤tBtBMtMtLtDtttDt
    • Local time (3)}{inf0tLsst ≥=>σInverse local time is a stable processwith index ½.
    • References R. Bass Probabilistic Techniques inAnalysis, Springer, 1995 F. Knight Essentials of Brownian Motionand Diffusion, AMS, 1981 I. Karatzas and S. Shreve BrownianMotion and Stochastic Calculus, Springer,1988