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 nu. 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,W-2(H,nu) and to obtain information on the domain of the infinitesimal generator of P-t.