Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces
type: Item
instance of: scholarly article
title: Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces
zbMATH Open document ID: 1170.47029
author: Abdelhadi Es-Sarhir
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-13T00:00:00Z
review text: This paper investigates the following semilinear stochastic differential equation on a separable Hilbert space: \[ d X(t)= (AX(t) + (-A)^{1/2} F(X(t))) d t + d W_t, \] where \(A\) is a negatively definite selfadjoint operator with compact resolvent, \(W\) is the cylindrical Brownian motion, and \(F\) is a global Lipschitz continuous function. It is proved that, if \[ \int_0^1s^{-\varepsilon}\|e^{sA}\|_{HS}^2 \,ds <\infty \] holds for some \(\varepsilon\in (0,1),\) then the equation has a unique mild solution and the corresponding semigroup is strong Feller. If, furthermore, \(\langle x-y, F(x)-F(y)\rangle\) holds for all \(x,y\), then the semigroup has a unique invariant probability measure. These extend some known results for equations without \((-A)^{1/2}\) in front of \(F\).
reviewed by: Feng-Yu Wang
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60B11
zbMATH DE Number: 5553631
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: strong Feller property
zbMATH Keywords: invariant measure
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.12.031
OpenAlex ID: W1991645574
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DOI: 10.1016/J.JMAA.2008.12.031
description: scientific article; zbMATH DE number 5553631