A solution theory for a general class of SPDEs

André Süß, Marcus Waurick

Research output: Contribution to journalArticle

4 Citations (Scopus)
27 Downloads (Pure)

Abstract

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of RdRd. The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green’s function is not required.
Original languageEnglish
Pages (from-to)278-318
Number of pages41
JournalStochastic and Partial Differential Equations: Analysis and Computations
Volume5
Issue number2
Early online date25 Nov 2016
DOIs
Publication statusPublished - 30 Jun 2017

Keywords

  • stochastic partial differential equations
  • evolutionary equations
  • stochastic equations of mathematical physics
  • weak solutions

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