Ergodic properties of recurrent solutions of stochastic evolution equations
type: Item
instance of: scholarly article
title: Ergodic properties of recurrent solutions of stochastic evolution equations
zbMATH Open document ID: 0820.60040
author: B. Maslowski
author: Jan Seidler
published in: Osaka Journal of Mathematics
publication date: 1995-09-11T00:00:00Z
review text: Ergodic and recurrence properties for a semilinear stochastic equation of the general form \[ dX(t) = (AX(t) + f(X(t)))dt + dW_ t, \tag{1} \] in a real separable Hilbert space \(H\) are investigated, where the linear operator \(A\) generates a strongly continuous semigroup on \(H\), \(W_ t\) is a cylindrical Wiener process on \(H\) with a bounded covariance operator and \(f\) is a nonlinear function defined on a subspace of \(H\). The aim of the paper is to study \(\sigma\)-finite and probabilistic invariant measures for the equation (1). Under suitable conditions (roughly speaking, sufficient nondegeneracy of the noise and a recurrence condition) the ratio ergodic theorem and some of its consequences as well as the existence and uniqueness of a \(\sigma\)-finite invariant measure are shown. In such cases, there is a dichotomy in behavior of the solutions to (1), analogous to the one in the finite-dimensional case.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 755432
zbMATH Keywords: recurrence properties
zbMATH Keywords: semilinear stochastic equation of the general form
zbMATH Keywords: invariant measures
zbMATH Keywords: ratio ergodic theorem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 755432