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Cubature methods for stochastic (partial) differential equations in weighted spaces [Elektronski vir]
Dörsek, Philipp ; Teichmann, Josef ; Velušček, Dejan
Razvijemo šibke aproksimacijske metode imenovane kubature na Wienerjevem prostoru za stohastične parcialne diferencialne enačbe (SPDE) pri neomejenih karakteristikah in neomejenih testnih funkcijah. ...Najprej opišemo nedavno predstavljeno fleskibilno funkcionalno analitično okolje: utežene prostore v katerih veljajo lastnosti podobne Fellerjevim. Podrobna analiza vektorskih polj na uteženih prostorih nam izda optimalno stopnjo kovergence kubaturnih metod za SPDE Da Prato-Zabczyk tipa. Dokažemo univerzalno stabilnost operatorja lokalne aproksimacije v izbranem funkcionanem analitičnem okolju za SPDE, pri čemer pa v neskončno razsežnem primeru potrebujemo novo tehnično predpostavko, to je šibko simetričnost kubaturnih formul. Prikažemo tudi računske rezultate za kubaturno diskretizacijo prostorsko razširjenega FitzHugh-Nagumo modela iz matematične biologije, ki lepo predstavijo uporabnost naše teorije.
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