Exponentially stable stationary solutions for stochastic evolution equations and their perturba\-tion
type: Item
instance of: scholarly article
title: Exponentially stable stationary solutions for stochastic evolution equations and their perturba\-tion
zbMATH Open document ID: 1066.60058
author: Peter E. Kloeden
author: T. Caraballo
author: Björn Schmalfuss
published in: Applied Mathematics and Optimization
publication date: 2005-02-21T00:00:00Z
review text: For the abstract evolution equation \(dX=\bigl(AX+f(X)\bigr)\,dt+B(X)\,dW_t\), where \(A\) generates a strongly continuous semigroup with growth bound \(a\) on a separable Hilbert space \(H\), \(f:H\to H\) is globally Lipschitz with Lipschitz constant \(L_f\), \(W\) is a trace class Wiener process on some Hilbert space \(U\), and \(B\) is globally Lipschitz with Lipschitz constant \(L_B\) from \(H\) to the space of linear operators from \(U\) to \(H\), it is assumed that \[ a+L_f+L_B/2<0.\tag{1} \] Under this condition, existence of an exponentially attracting stationary solution is proved. Assuming a finite-dimensional Wiener process with \(B\) linear and commuting with \(A\), it is shown that exponential attraction holds uniformly in bounded sets of initial conditions. Finally, it is shown that the stationary solution depends continuously on perturbations of the nonlinearity \(f\), provided (1) holds uniformly.
reviewed by: Hans Crauel
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 93E15
zbMATH DE Number: 2135339
zbMATH Keywords: random dynamical systems
zbMATH Keywords: random attractors
MaRDI profile type: Publication
full work available at URL: s00245-004-0802-1
OpenAlex ID: W1999004431
DOI: 10.1007/S00245-004-0802-1
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description: scientific article; zbMATH DE number 2135339