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1、ANINTRODUCTIONTOSTOCHASTICPARTIALDIFFERENTIALEQUATIONSJohnB.WALSHThegeneralproblemisthis.Supposeoneisgivenaphysicalsystemgovernedbyapartialdifferentialequation.Supposethatthesystemisthenperturbedrandomly,perhapsbysomesortofawhitenoise.Howdoesitevolveintime?Thinkforexampleofaguitarcar
2、elesslyleftoutdoors.Ifu(x,t)isthepositionofoneofthestringsatthepointxandtimet,thenincalmairu(x,t)wouldsatisfythewaveequationu=u.However,ifattxxsandstormshouldblowup,thestringwouldbebombardedbyasuccessionofsandgrains.LetWrepresenttheintensityofthebombardmentatthepointxtxandtimet.Thenu
3、mberofgrainshittingthestringatagivenpointandtimewillbelargelyindependentofthenumberhittingatanotherpointandtime,sothat,aftersubtractingameanintensity,Wmaybeapproximatedbyawhitenoise,andthefinalequationisUtt(x,t)=Uxx(X,t)+W(x,t),whereWisawhitenoiseinbothtimeandspace,or,inotherwords,at
4、wo-parameterwhitenoise.Onepeculiarityofthisequation-notsurprisinginviewofthebehaviorofordinarystochasticdifferentialequations-isthatnoneofthepartialderivativesinitexist.However,onemayrewriteitasanintegralequation,andthenshowthatinthisformthereisasolutionwhichisacontinuous,thoughnon-d
5、ifferentiable,function.Inhigherdimensions-withadrumhead,say,ratherthanastring-eventhisfails:thesolutionturnsouttobeadistribution,notafunction.Thisisoneofthetechnicalbarriersinthesubject:onemustdealwithdistribution-valuedsolutions,andthishasgeneratedanumberofapproaches,mostinvolvingaf
6、airlyextensiveuseoffunctionalanalysis.267Ouraimistostudyacertainnumberofsuchstochasticpartialdifferentialequations,toseehowtheyarise,toseehowtheirsolutionsbehave,andtoexaminesometechniquesofsolution.Weshallconcentratemoreonparabolicequationsthanonhyperbolicorelliptic,andonequationsin
7、whichtheperturbationcomesfromsomethingakintowhitenoise.Inparticular,oneclassweshallstudyindetailarisesfromsystemsofbranchingdiffusions.TheseleadtolinearparabolicequationswhosesolutionsaregeneralizedOrnstein-Uhlenbeckprocesses,andincludethosestudiedbyIto,HolleyandStoock,Dawson,andothe
8、rs.Anotherre
