A Trotter-type theorem for nonlinear stochastic equations in variational formulation and homogenization

Abstract

This paper is concerned with the nonlinear partial differential equations of calculus of variations perturbed by noise in the Gelfand triple $V\subset H\subset V^{\prime }$. The main result is a Trotter-type theorem for this equation. In the second part of the paper we prove that, if we assume graph convergence of the sequence of nonlinear operators $ \{ A^{\alpha} \} _{\alpha }$, we have convergence of the corresponding sequence of invariant measures. Those results are used in the last part of the paper to study the homogenization problem for the equation, in the case of the differential operator of the type $A ( u ) =- \text{\mathrm {div}} [ a ( \nabla u ) ], $ for $u\in H_{0}^{1} ( \mathcal{O} ).$