Exponential stability of non-linear stochastic evolution equations
type: Item
instance of: scholarly article
title: Exponential stability of non-linear stochastic evolution equations
zbMATH Open document ID: 0933.60072
DOI: 10.1016/S0304-4149(98)00048-9
author: Kai Liu
author: Xuerong Mao
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: Let \(K,H\) be real separable Hilbert spaces, \(V\) a uniformly rotund Banach space with a uniformly rotund dual space \(V^*\), let \(V\subseteq H\subseteq V^*\) with continuous dense embeddings. Let \(W\) be a Wiener process in \(K\) with a nuclear covariance operator, defined on a filtered probability space \((\Omega , \mathcal F, (\mathcal F_{t}),P)\). A stochastic differential equation \[ X_{t} = x_{0} + \int ^{t}_{0} f(t,X_{s}) ds + \int ^{t}_{0} g(s,X_{s}) dW_{s}, \quad t\geq 0, \tag{1} \] in \(V^*\) is considered, where \(f(t,\cdot)\) is a family of nonlinear operators from \(V\) into \(V^*\) and \(g(t,\cdot):V\to \mathcal L(K,H)\) are locally bounded mappings. Assume that there exists a unique strong solution \(X_{t}(x_{0})\in L^{2}((0,T)\times \Omega ;V) \cap L^{2}(\Omega ;\mathcal C([0,T];H))\) for any \(T>0\) and any \(\mathcal F_0\)-measurable \(V\)-valued random function \(x_{0}\) with \(E\|x_0\|^{2}_{V}<\infty \). Sufficient conditions for the equation (1) to be almost surely exponentially stable (that is, for the existence of a \(\gamma >0\) such that \(\limsup_{t\to \infty} t^{-1}\log \|X_t(x_0)\|_{H} \leq -\gamma \) almost surely for any initial condition \(x_{0}\)) are given, their proofs being based on Lyapunov functions techniques. Analogous results are established also for stochastic evolution equations with time delays.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 93E03
zbMATH DE Number: 1364520
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: stochastic evolution equations with delays
zbMATH Keywords: almost sure exponential stability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1364520