微分方程和随机微分方程

March5,2013ODE_x=V(x);x2­½RnV:­!Rnx(t)t¸0tSDEODESDEdX(t)=V(X(t))dt+G(X(t))»(t);t0G(X):n£m,»(¢):=m-dimensionalwhitenoise:²»(¢)²X(¢)solvesSDE1.X(0)=0,m=n,V´0,G´I,BrownW(¢),WienerdW(¢)=»(¢)Brown2.It^o-n=1,u:R1!R1,X(¢)SDE,Y(t):=u(X(t))dY=u0dX=u0(Vdt+GdW)wrong!It^odW¼(dt)1=2insomesensedY=u0dX+12u00(dX)2+¢¢¢=u0(Vdt+GdW)+12u00(Vdt+GdW)2+¢¢¢=³u0V+12G2u00´dt+u0GdW+¢¢¢dY=³u0V+12G2u00´dt+u0GdW(­;U;P)­U­¾-P:U![0;1]UA2U;!2­P(A)ABRnBorelf¸0,Rfdx=1,P(B)=ZBfdx8B2B:(Rn;B;P)fP(­;U;P)(1)X:­!RnU¡X¡1(B)2U8B2B(2)fX(t)jt¸0g.(3)!2­,t7!X(t;!):E(X):=Z­XdP:V(X):=Z­jX¡E(X)j2dPfXP(X2B)=ZBf(x)dx8B2BX:­!R1f(x)=1p2¼¾2e¡jx¡mj22¾2x2R1X()N(m;¾2)BrownWienerR.Brown(1826-27)L.Bachelier(1900)A.Einstein(1905)7!N.Wiener(1920’s)BrownWiener(n=1)W(¢)BrownWiener(i)W(0)=0a.s.,(ii)W(t)¡W(s)N(0;t¡s),8t¸s¸0,(iii)0t1¢¢¢tk,W(t1),W(t2)¡W(t1);¢¢¢;W(tk)¡W(tk¡1)HÄolderKolmogorovTheorem:X(¢)®0,¯0,E(jX(t)¡X(s)j¯)·Cjt¡sj1+®:0°®=¯T0!X(¢;!)[0;T]°¡HÄolderBrownTheorem°1=2!W(¢;!)°¡HÄolderTheorem1=2·°·1!W(¢;!)°¡HÄolderTheoremBrownW(¢)MarkovIt^oRt0GdWStochasticprocessXsolvesSDEdX(t)=V(X(t))dt+G(X(t))dW(t);t0ifX(t)=X(0)+Zt0V(X(s))ds+Zt0G(X(s))dW(s)a:s:ObservationZt0WdW=?RiemannSums0=t0t1¢¢¢tm=t;¿k:=(1¡¸)tk+¸tk+1§m¡1k=0W(¿k)(W(tk+1)¡W(tk))!W(t)22+(¸¡12)tinL2(­),asmaxkjtk+1¡tkj!0.¿k!It^o¸=0ZtsWdW=W(t)2¡W(s)22¡t¡s2:Whataretheadvantagesoftaking¸=0?²It^o:7!generatorL¤²²²°¡HÄolder(for°1=2)a.s.²StratonovichRG(X)±dW²PDEPDEU½Rnu½¡124u=1inUu=0on@U:x2U,X(¢)xBrown,X(¢)=W(¢)+x:¿x:=firsttimeX(¢)hits@UTheorem(Feynman-Kac):u(x)=E(¿x)8x2U.BrowngeneratorL¤=124V=0G=I(i)Xs;x[s;T)X(s)=xP(s;x;t;B)=PrfXs;x(t)2Bg;B½Btransitionprobability(ii)P(s;x;t;B)=ZBp(s;x;t;y)dy;pFokker-Planck(kernel).²Probabilityapproach:¼t:M(Rn)!M(Rn):¼t¹(B)=ZRnP(0;x;t;B)¹(dx):–Restrictionsareneededonnotonlynoisebutalsodynamics.²PDEapproach:Fokker-Planck,non-LipschizV(e.g.integrable)andlessregularGModeling&DynamicsIssues²Modelingissues:–Noiseiseverywhere.Sowhencanoneusedeterministicmodels?(Overafinitetimeinterval,smallnoisecanoftenbeneglected)–Whendoesnoiseimpactbecometrulyessential?(Overaninfinitetimeinterval,noisecanbesignificant)²Generaldynamicsissues:stochasticstability:Classify“dynamicswhichare“robustunderthenoiseperturbations.–Generaldynamicssubjectsareinvariantmeasures(andinvariantsets)ofODE.Fokker-Planck8:ut=Lu=:nPi;j=1@2ij(aiju)¡div(Vu);x2Rn;t0;u(x;t)¸0;RRnu(x;t)dx=1;where(aij)=GG2:Assume(aij)0everywhere.²StationaryFokker-Planckequation:Undersomeconditions,u(x;t)!u(x)ast!+1,andu(x)satisfies8:Lu=:nPi;j=1@2ij(aiju)¡div(Vu)=0;x2Rnu(x)¸0;RRnu(x)dx=1:SteadyStates²measuresolution:Aprobabilitymeasure¹onRns.t.ZRnL¤fd¹=0;8f2C10(Rn)whereL¤f:=nPi;j=1aij@2ijf+V¢rfistheadjointFPoperator.²Weaksolution:Apositivefunctionu2W1;ploc(pn)s.t.¹(dx)=udx,i.e.uisthedensityfunctionofameasuresolution.–Whencoefficientsaresmooth,ubecomesaclassicalsolution.²RegularityTheorem(Bogachev-Krylov-Röckner,01):Assumeaij2W1;ploc;V2Lploc(pn).Thenmeasuresolutionsareweaksolutions.Existence1)Thecaseofcompactmanifolds.WithV;Gsmooth,theFPequationdefinedonacompactmanifoldadmitsauniquestrongstationarysolution(Zeeman,1988).2)ThecaseofboundeddomainsinRnwithLuni.elliptic.ExistenceofweakstationarysolutionsfollowsfromclassicalworkofTrudinger(1973,fornon-homogeneousDirichletboundarycondition)andAmann(1983,forhomogeneousRobinboundarycondition).–Existenceinanunboundeddomainrequiresadditionalconditions.Forinstance,nosolutionexistsfor½4u=0;x2Rn;u(x)¸0;RRnu(x)dx=1:ResultsinRnKnownresultsinRnassumetheexistenceofastrongLyapunovfunctionU2C2(Rn),i.e.,i)limx!1U(x)=+1;ii)limx!1L¤U(x)=¡1:²Khasminskii’sTheorem(1960,1980):AssumethatV;aijarelocallyLipschizinRnandthat9astrongLyapunovfunctionU.ThentheFPeq.admitsauniquestationarysolution.²Extensionstonon-Lipcoefficients:Bensoussan(1988),Skorohod(1989),Veretennikov(1987,1997,1999),Albeverio,Bogachev,Krylov,Röckner,Stannat(1997-2002),...²Theorem(2001):Assumea)aij2W1;ploc;V2Lplocforsomepn;b)9astrongLyapunovfunctionU.ThentheFPeq.onRnadmitsauniquestationarymeasurewithdensitylyinginW1;ploc.²Uniqueness:Varadham(1980),Albeverio-Bogachev-Röckner(1999)—Counterexamplesonuniqueness:Bogachev-Röckner-Stannat(1999,2002),Shaposhnikov(2008)²Fordegenerate(aij):TheabovetheoremisvalidinthiscasewhenV2C0,butthemeasuresolutionsneednotadmitdensityfunctionsandneednotbeunique.GeneralExistence&Non-existenceResultCollaborators:W.Huang(USTC)Z.Liu(JilinU.)YingfeiYi(Geo.Tech.)Weassumetheexistenceofa(weaker)LyapunovfunctionU2C2(Rn),i.e.,9º0s.t.i)limx!1U(x)=+1;ii)limx!1L¤U(x)·¡º:²Theorem:Assumea)andthat9aLyapunovfunctionU.ThentheFPeq.admitsauniquestationarymeasurewithdensitylyinginW1;ploc.²Theorem(Non-existence):Assumea)andthat9U2C2(Rn)s.t.i)limx!1U(x)=+1,limx!1L¤U(x)¸º0;ii)9positivefunctionHwithR11H¡1(t)dt=+1s.t.0nXi;j=1aij@xiU@xjU·H(U);8jxjÀ1:ThentheFPeq.admitsnostationarymeasures.–Usingtheexistenceandnon-existenceresults,onecanobtainanecessaryandsufficientconditionfortheexistenceofstationarymeasuresinthecaseofsmallnoise.–Theexistencepartofthetheoremalsoholdsfordegenerate(aij).–Boththeoremscanbealsoshownrespectivelyforsomecaseswithº=0.–Boththeorem