Appl Math Optim 54:401–415 (2006)
2006 Springer Science+Business Media, Inc.
The Pathwise Numerical Approximation of Stationary Solutions
of Semilinear Stochastic Evolution Equations
and P. E. Kloeden
Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla,
Apdo. de Correos 1160, 41080 Sevilla, Spain
Fachbereich Mathematik, Johann Wolfgang Goethe Universit¨at,
D-60054 Frankfurt am Main, Germany
Abstract. Under a one-sided dissipative Lipschitz condition on its drift, a stochas-
tic evolution equation with additive noise of the reaction–diffusion type is shown to
have a unique stochastic stationary solution which pathwise attracts all other solu-
tions. 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
Key Words. Stochastic partial differential equations, Random partial and ordinary
differential equations, Galerkin approximations, Implicit Euler scheme, Stationary
solutions, Ornstein–Uhlenbeck solution.
AMS Classiﬁcation. Primary 60H35, Secondary 60H15, 60H25.
This work was partially supported by the Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER
(European Community) Grant BFM2002-03068. Peter Kloeden was also supported by the Ministerio de
Educaci´on y Ciencia (Spain) under Grant SAB2004-0146, within the Programa de Movilidad del Profesorado
universitario espa˜nol y extranjero.