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 Pt. 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 W1,2(H,ν) and to obtain information on the domain of the infinitesimal generator of Pt.
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Barbu, V., Da Prato, G. Ergodicity for Nonlinear Stochastic Equations in Variational Formulation. Appl Math Optim 53, 121–139 (2006). https://doi.org/10.1007/s00245-005-0838-x
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DOI: https://doi.org/10.1007/s00245-005-0838-x