Ergodicity for Nonlinear Stochastic Equations in Variational Formulation

Ergodicity for Nonlinear Stochastic Equations in Variational Formulation This paper is concerned with nonlinear partial differential equations of the calculus of variation (see (13)) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see (14)), using monotonicity methods. The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P t . We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W 1,2 (H,ν) and to obtain information on the domain of the infinitesimal generator of P t . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Ergodicity for Nonlinear Stochastic Equations in Variational Formulation

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Publisher
Springer-Verlag
Copyright
Copyright © 2006 by Springer
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-005-0838-x
Publisher site
See Article on Publisher Site

Abstract

This paper is concerned with nonlinear partial differential equations of the calculus of variation (see (13)) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see (14)), using monotonicity methods. The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P t . We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W 1,2 (H,ν) and to obtain information on the domain of the infinitesimal generator of P t .

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 1, 2006

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