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) Invarian...
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 processe...
Markov processes}0,|,{}|,{ suBstBBstB utst ≤≤≥=≥ LLThe theory of Markov processes usestools from several branches of analy...
Martingales are the only family of processesfor which the theory of stochastic integrals isfully developed, successful and...
Gaussian processesnttt BBB ,,, 21 is multidimensionalnormal (Gaussian)(i) Excellent bounds for tails(ii) Second moment ca...
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,...
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:dxxud...
Multidimensional Brownian motion,,, 321ttt BBB - independent 1-dimensionalBrownian motions),,,( 21 dttt BBB  - d-dimensi...
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 i...
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}},{,)|(0tsB...
The Ito formulaDisappearing terms (2)∫∫∫∂∂+∂∂+∂∂=−tststsstdsBsfxdsBsfsdBBsfxBtfBtf022000),(21),(),(),(),(∫∫ ∂∂+∂∂=−tststds...
Mild modifications of BMMild = the new process correspondsto the Laplacian(i) Killing – Dirichlet problem(ii) Reflection –...
Related models – diffusionsdtXdBXdX tttt )()( µσ +=(i) Markov property – yes(ii) Martingale – only if(iii) Gaussian – no, ...
Related models – stable processes2/1)(dtdB =α/1)(dtdX =Brownian motion –Stable processes –(i) Markov property – yes(ii) Ma...
Related models – fractional BMα/1)(dtdX =(i) Markov property – no(ii) Martingale – no(iii) Gaussian – yes(iv) Continuous∞<...
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 propertie...
Exceptional points1|log|log2suplim0=↓ ttBtt1|)log(|log)(2suplim =−−−↓ ststBB ststFor any fixed s>0, a.s.,),0()(2suplim ∞∈−...
Cut pointsFor any fixed t>0, a.s., the 2-dimensionalBrownian path contains a closed looparound in every intervaltB ),( ε+t...
Intersection properties1))((.,.)4(2))((.,.2))((.,.)3())((.,...)2(.,.)1(14131312≤∈∀=≤∈∀=∈∃=∞=∈∃≠≠∀∀==≠∃∀=−−−−xBCardRxsadxBC...
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 ref...
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 Diffus...
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Brownian motion by krzysztof burdzy(university of washington)

  1. 1. BROWNIAN MOTIONA tutorialKrzysztof BurdzyUniversity of Washington
  2. 2. A paradox∞<′′→∈|)(|sup,]1,0[:]1,0[tfRft(*)))((21exp)()10,)()((102′−≈<<+<<−∫ dttfcttfBtfP tεεε
  3. 3. (*) is maximized by f(t) = 0, t>0Microsoft stock- the last 5 yearsThe most likely (?!?) shape of aBrownian path:
  4. 4. Definition of Brownian motionBrownian motion is the unique processwith the following properties:(i) No memory(ii) Invariance(iii) Continuity(iv) tBVarBEB tt === )(,0)(,00
  5. 5. Memoryless process0t2t1t 3t,,, 231201 tttttt BBBBBB −−−are independent
  6. 6. InvarianceThe distribution ofdepends only on t.sst BB −+
  7. 7. Path regularity(i) is continuous a.s.(ii) is nowhere differentiable a.s.tBt →tBt →
  8. 8. Why Brownian motion?Brownian motion belongs to several familiesof well understood stochastic processes:(i) Markov processes(ii) Martingales(iii) Gaussian processes(iv) Levy processes
  9. 9. 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)
  10. 10. Martingales are the only family of processesfor which the theory of stochastic integrals isfully developed, successful and satisfactory.Martingalessst BBBEts =⇒< )|(∫tss dBX0
  11. 11. Gaussian processesnttt BBB ,,, 21 is multidimensionalnormal (Gaussian)(i) Excellent bounds for tails(ii) Second moment calculations(iii) Extensions to unordered parameter(s)
  12. 12. The Ito formula∑∫ =+∞→−=ntknknknkntss BBXdBX0//)1(/0)(lim∫ ∫ ′′+′+=t tssst dsBfdBBfBfBf0 00 )(21)()()(
  13. 13. Random walkIndependent steps, P(up)=P(down){ } { }0,0,/ ≥ →≥ ∞→tBtWa taat(in distribution)tWt
  14. 14. ScalingCentral Limit Theorem (CLT),parabolic PDE’s}10,{}10,{ / ≤≤=≤≤ tBatB atDt
  15. 15. 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
  16. 16. Invariance (2)}10,{}10,{ 11 ≤≤−=≤≤ − tBBtB tDtTime reversal0 1
  17. 17. 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
  18. 18. Multidimensional Brownian motion,,, 321ttt BBB - independent 1-dimensionalBrownian motions),,,( 21 dttt BBB  - d-dimensional Brownianmotion
  19. 19. Feynman-Kac formula (2))(xf−= ∫ττ0)(exp)()( dsBVBfExu sxτB x0)()()(21=−∆ xuxVxu
  20. 20. 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
  21. 21. Conformal invariancef}0),()({ 0 ≥− tBfBf tanalytictB )( tBfhas the same distribution as∫ ′=≥tstc dsBftctB02)( |)(|)(},0,{
  22. 22. If thenThe Ito formulaDisappearing terms (1)∫ ∫∆+∇+=t tssst dsBfdBBfBfBf0 00 )(21)()()(∫∇+=tsst dBBfBfBf00 )()()(0≡∆f
  23. 23. Brownian martingalesTheorem (Martingale representation theorem).{Brownian martingales} = {stochastic integrals}},{,)|(0tsBFMMMMEdBXMsBttssttsst≤=∈== ∫σ
  24. 24. The Ito formulaDisappearing terms (2)∫∫∫∂∂+∂∂+∂∂=−tststsstdsBsfxdsBsfsdBBsfxBtfBtf022000),(21),(),(),(),(∫∫ ∂∂+∂∂=−tststdsBsfxEdsBsfsEBtEfBtEf02200),(21),(),(),(
  25. 25. Mild modifications of BMMild = the new process correspondsto the Laplacian(i) Killing – Dirichlet problem(ii) Reflection – Neumann problem(iii) Absorption – Robin problem
  26. 26. Related models – diffusionsdtXdBXdX tttt )()( µσ +=(i) Markov property – yes(ii) Martingale – only if(iii) Gaussian – no, but Gaussian tails0≡µ
  27. 27. 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 ≤<
  28. 28. Related models – fractional BMα/1)(dtdX =(i) Markov property – no(ii) Martingale – no(iii) Gaussian – yes(iv) Continuous∞<<α1∞<< 21
  29. 29. Related models – super BMSuper Brownian motion is related to2uu =∆and to a stochastic PDE.
  30. 30. Related models – SLESchramm-Loewner Evolution is a modelfor non-crossing conformally invariant2-dimensional paths.
  31. 31. (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
  32. 32. 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
  33. 33. 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
  34. 34. Intersection properties1))((.,.)4(2))((.,.2))((.,.)3())((.,...)2(.,.)1(14131312≤∈∀=≤∈∀=∈∃=∞=∈∃≠≠∀∀==≠∃∀=−−−−xBCardRxsadxBCardRxsaxBCardRxsadxBCardRxsaBBtssatdBBtstsadtsts
  35. 35. Intersections of random sets∅≠∩>+BAdBA)dim()dim(The dimension of Brownian trace is 2in every dimension.
  36. 36. 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
  37. 37. Local timeststBtBMdsL s≤<<−→== ∫sup21lim }{0t01 εεε ε
  38. 38. Local time (2)}10,|{|}10,{}10,{}10,{≤≤=≤≤−≤≤=≤≤tBtBMtMtLtDtttDt
  39. 39. Local time (3)}{inf0tLsst ≥=>σInverse local time is a stable processwith index ½.
  40. 40. 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
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