A Variational Approach to Stochastic Nonlinear Diffusion Problems with Dynamical Boundary Conditions

We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions which depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; further, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.