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1、StochasticPartialDifferentialEquationsLecturesgiveninFudanUniversity,Shangha¨ı,April2007E.Pardoux´Marseille,France2Contents1IntroductionandMotivation51.1Introduction............................51.2Motivation.............................61.2.1Turbulence........................71.2.2Populationd
2、ynamics,populationgenetics.......71.2.3Neurophysiology.....................81.2.4Evolutionofthecurveofinterestrate..........81.2.5NonLinearFiltering...................81.2.6Movementbymeancurvatureinrandomenvironment.91.2.7Hydrodynamiclimitofparticlesystems.........101.2.8Fluctuationsofaninte
3、rfaceonawall..........112SPDEsasinfinitedimensionalSDEs132.1ItˆocalculusinHilbertspace...................132.2SPDEwithadditivenoise....................162.2.1Thesemi–groupapproachtolinearparabolicPDEs..172.2.2ThevariationalapproachtolinearandnonlinearparabolicPDEs......................192.3Va
4、riationalapproachtoSPDEs.................252.3.1Monotone–coerciveSPDEs...............252.3.2Examples.........................352.3.3CoerciveSPDEswithcompactness...........372.4SemilinearSPDEs.........................433SPDEsdrivenbyspace–timewhitenoise493.1Restrictiontoone–dimensionalspaceva
5、riable.........493.2Ageneralexistence–uniquenessresult..............513.3Moregeneralexistenceanduniquenessresult..........5934CONTENTS3.4Positivityofthesolution.....................593.5ApplicationsofMalliavincalculustoSPDEs..........603.6SPDEsandthesuperBrownianmotion.............663.6.1The
6、caseγ=1/2.....................663.6.2Othervaluesofγ<1...................733.7SPDEswithsingulardrift,andreflectedSPDEs........793.7.1ReflectedSPDE......................803.7.2SPDEwithcriticalsingulardrift............82Chapter1IntroductionandMotivation1.1IntroductionWeshallstudyintheselecturespa
7、rabolicPDEs,whichwillbemostlynonlinear.Thegeneraltypeofequationswehaveinmindisoftheform∂u·2(t,x)=F(t,x,u(t,x),Du(t,x),Du(t,x))+G(t,x,u(t,x),Du(t,x))W(t,x),∂torinthesemilinearcase∂u·(t,x)=∆u+f(t,x,u(t,x))+g(t,x,u(t,x))W(t,x).∂t·Weshallmakeprecisewha
