Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations

Yuri Bakhtin, Jonathan C. Mattingly

Research output: Contribution to journalArticle

Abstract

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.

Original languageEnglish (US)
Pages (from-to)553-582
Number of pages30
JournalCommunications in Contemporary Mathematics
Volume7
Issue number5
DOIs
StatePublished - Oct 2005

Keywords

  • Ergodicity
  • Lyapunov functions
  • Memory
  • Stationary solutions
  • Stochastic Ginsburg-Landau equation
  • Stochastic Navier-Stokes equation
  • Stochastic differential equations

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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