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.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Caraballo, T., Kloeden, P. The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations. Appl Math Optim 54, 401–415 (2006). https://doi.org/10.1007/s00245-006-0876-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-006-0876-z