Transition semigroups corresponding to Lipschitz dissipative systems.
type: Item
instance of: scholarly article
title: Transition semigroups corresponding to Lipschitz dissipative systems.
zbMATH Open document ID: 1062.47043
author: Guiseppe Da Prato
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-07-01T00:00:00Z
review text: Consider the following stochastic differential equation on a separable Hilbert space \(H\): \[ d X_t = (A X_t +F(X_t))d t + dW_t, \] where \(W_t\) is the cylindrical Wiener process, \((A, D(A))\) the generator of a \(C_0\)-semigroup in \(H\) of negative type, and \(F\) a Lipschitz dissipative mapping on \(H\). Let \(P_t\) be the transition semigroup associated to the mild solution which has a unique invariant measure \(\nu\). Then under certain conditions, the infinitesimal generator \(N_p\) of \(P_t\) on \(L^p(\nu)\), \(p\geq 1,\) coincides with the \(L^p\)-closure of the following operator defined on the space of exponential functions: \[ N_0 f(x):= \frac 1 2 \text{Tr}(D^2 f(x)) +\langle x,A^*Df(x)\rangle +\langle F(x), Df(x)\rangle,\;\;x\in H. \] Moreover, it is proved that \(\nu\) is absolutely continuous with respect to the invariant measure associated to the equation with \(F=0.\)
reviewed by: Feng-Yu Wang
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2077713
zbMATH Keywords: Markov semigroup
zbMATH Keywords: transition semigroup
zbMATH Keywords: dissipative system
zbMATH Keywords: stochastic differential equation
MaRDI profile type: Publication
full work available at URL: dcds.2004.10.177
OpenAlex ID: W2082612988
DOI: 10.3934/DCDS.2004.10.177
description: scientific article; zbMATH DE number 2077713