Invariant measures for dichotomous stochastic differential equations in Hilbert spaces
type: Item
instance of: scholarly article
title: Invariant measures for dichotomous stochastic differential equations in Hilbert spaces
zbMATH Open document ID: 1002.60052
DOI: 10.1016/S1631-073X(02)02410-X
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-01-01T00:00:00Z
review text: The authors are interested in the existence of invariant measures for semilinear stochastic evolution equations in a real separable Hilbert space \(H\) \[ dX(t)=\{AX(t)+F[X(t)]\} dt+\sum_{i=0}^\infty\Phi_i[X(t)] dW_i(t),\quad X(0)=X_0, \quad t\geqslant 0,\tag{1} \] where \(A\) is the generator of a \(C_0\)-semigroup \((S(t))_{t\geqslant 0}\) on \(H\), the maps \(F\) and \(\Phi_i\), \(i\in\mathbb N\), are Lipschitz continuous from \(H\) to \(H\), and \(\{W_i\}_{i=0}^\infty\) is a sequence of independent scalar Wiener processes. Mild solutions being functions from \([0,\infty)\) to the Bochner space \(L^2(\Omega; H)\) are considered, where \((\Omega,\mathcal F,\mathbb P)\) is the underlying probability space. Assuming that the nonlinearities \(F\) and \(\Phi_i\) are Lipschitz continuous, infinite-dimensional noise that is white in time and colored in space is considered. It is shown that the existence of a bounded solution of (1) implies existence of an invariant measure of (1) if the equation (1) has an exponential dichotomy in the sense that the semigroup \((S(t))_{t\geqslant 0}\) is hyperbolic and the Lipschitz constants of \(F\) and \(\Phi_i\) are not too large.
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1777422
zbMATH Keywords: semilinear stochastic evolution equations
zbMATH Keywords: real separable Hilbert space
zbMATH Keywords: Wiener process
zbMATH Keywords: invariant measure
zbMATH Keywords: mild solutions
zbMATH Keywords: dichotomous equation
zbMATH Keywords: infinite-dimensional noise
zbMATH Keywords: white noise
zbMATH Keywords: colored noise
zbMATH Keywords: hyperbolic semigroup
zbMATH Keywords: Lipschitz constants
author: O. W. van Gaans
author: Sjoerd M. Verduyn Lunel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1777422