Long-time behaviour of nonautonomous SPDE's.
type: Item
instance of: scholarly article
title: Long-time behaviour of nonautonomous SPDE's.
zbMATH Open document ID: 1058.60051
DOI: 10.1016/S0304-4149(01)00105-3
author: B. Maslowski
author: Isabel Simão
published in: Stochastic Processes and their Applications
publication date: 2005-02-25T00:00:00Z
review text: The asymptotic behaviour of the time-inhomogeneous Markov process \((X,\mathbf P_ {s,x})\) defined by a stochastic evolution equation \(dX =\{ AX + f(t,X)\}\,dt + Q^ {1/2}\,dW\) in a Hilbert space \(H\) is studied. Here, \(W\) is a standard cylindrical Wiener process on \(H\), \(Q\) is a nonnegative self-adjoint bounded linear operator in \(H\), \(A\) is an infinitesimal generator of a strongly continuous semigroup on \(H\) and \(f:\mathbb R_ {+}\times H\to H\) is a Borel mapping. Define \(P_ {s,t} \psi (x) = \mathbf E_ {s,x}\psi (X_ {t})\) for all \(t\geq s\geq 0\), \(x\in H\) and all bounded Borel real functions \(\psi \) on \(H\), denote by \(P_ {s,t}^ {*}\) the adjoint operators on the space of finite (signed) Borel measures on \(H\) and by \(\| \cdot \| _ {\text{var}}\) the total variation norm on this space. The main objective of the paper is to find sufficient conditions for \(\lim _ {t\to \infty }\| P^ {*}_ {s,t}\mu - P^ {*}_ {s,t}\varrho \| _ {\text{var}} =0\) to hold for any \(s\geq 0\) and all probability measures \(\mu \), \(\varrho \) on \(H\). Some interesting examples of stochastic second-order parabolic equations are covered by the obtained result. The proof is based on the method of lower measures; lower bounds for the Radon-Nikodym derivative of the transition function \(P(t,x,t+1,\cdot )\) with respect to a suitable Gaussian measure, which might be of independent interest, are derived in the course of the proof.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 2138899
zbMATH Keywords: nonautonomous stochastic partial differential equations
zbMATH Keywords: Markov processes
zbMATH Keywords: Girsanov theorem
MaRDI profile type: Publication
cites work: Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces
cites work: Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case
cites work: Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case
cites work: Stochastic Cahn-Hilliard equation
cites work: Strong feller property for stochastic semilinear equations
cites work: Stochastic burgers equation with correlated noise
cites work: Stochastic Equations in Infinite Dimensions
cites work: Ergodicity for Infinite Dimensional Systems
cites work: Ergodic results for stochastic navier-stokes equation
cites work: Ergodicity of the 2-D Navier-Stokes equation under random perturbations
cites work: Smoothing properties of nonlinear stochastic equations in Hilbert spaces
cites work: On invariant measures for diffusions on Banach spaces
cites work: Q5556832
cites work: Continuity of \(\ell ^ 2\)-valued Ornstein-Uhlenbeck processes
cites work: Ergodicity of stochastic plates
cites work: Q4853153
cites work: Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations
cites work: Chaos, fractals, and noise: Stochastic aspects of dynamics.
cites work: Stochastic evolution equations with random generators
cites work: Qualitative behaviour of solutions of stochastic reaction-diffusion equations
cites work: Q4204903
cites work: On probability distributions of solutions of semilinear stochastic evolution equations
cites work: Q4844847
cites work: Probabilistic approach to the strong Feller property
cites work: Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures
cites work: Coupling and invariant measures for the heat equation with noise
cites work: Evolution equation of a stochastic semigroup with white-noise drift.
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Strong Feller property and irreducibility for diffusions on Hilbert spaces
cites work: Ergodic behaviour of stochastic parabolic equations
cites work: Da Prato-Zabczyk's maximal inequality revisited. I.
cites work: Geometric ergodicity for stochastic pdes
full work available at URL: s0304-4149(01)00105-3
OpenAlex ID: W2010821222
description: scientific article; zbMATH DE number 2138899