Advertisement

Applied Mathematics and Optimization

, Volume 54, Issue 3, pp 401–415 | Cite as

The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations

  • T. CaraballoEmail author
  • P.E. KloedenEmail author
Article

Abstract

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions.

Keywords

Stationary Solution Galerkin Approximation Stochastic Partial Differential Equation Uhlenbeck Process Random Dynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Departamento de Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla, Apdo. de Correos 1160, 41080SevillaSpain
  2. 2.Fachbereich Mathematik, Johann Wolfgang Goethe Universitat, D-60054Frankfurt am MainGermany

Personalised recommendations