Stabilization of stationary solutions of evolution equations by noise
type: Item
instance of: scholarly article
title: Stabilization of stationary solutions of evolution equations by noise
zbMATH Open document ID: 1117.60063
author: Peter E. Kloeden
author: Björn Schmalfuss
author: T. Caraballo
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2007-02-12T00:00:00Z
review text: Existence, uniqueness, and exponential stability of non-constant stationary solutions of some semilinear evolution equation are studied. The equation under consideration is a mild formulation corresponding to an SPDE of the type \[ dX=AX\;dt+F(X)\; dt +B(X)\; dW \] on some separable Hilbert space, where \(A\) generates a \(C_0\)-semigroup, \(F\) and \(B\) are Lipschitz and \(W\) is some Wiener process. The main results show under some assumptions on \(B\) a stabilization effect on the deterministic dynamics. Here an exponentially stable solution for the stochastic equation exists, even if the equation without noise does not have any exponentially stable solution. The main tools in the proof are pullback techniques for random dynamical systems combined with exponential martingale inequalities. The stabilization relies on Ito-Stratonovic corrections.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35K40
zbMATH DE Number: 5124798
zbMATH Keywords: exponential stability
zbMATH Keywords: stationary solutions
zbMATH Keywords: SPDE
zbMATH Keywords: random dynamical system
zbMATH Keywords: mild solution
zbMATH Keywords: stabilization
MaRDI profile type: Publication
full work available at URL: dcdsb.2006.6.1199
OpenAlex ID: W2332877299
DOI: 10.3934/DCDSB.2006.6.1199
description: scientific article; zbMATH DE number 5124798