Mosco convergence of gradient forms with non-convex interaction potential

Abstract

Convexity of interaction potentials is a typical condition for the derivation of gradient bounds for diffusion semigroups in stochastic interface models, particle systems, etc. Gradient bounds are often used to show convergence of semigroups. However for a large class of convergence problems the assumption of convexity fails. The article suggests a way to overcome this hindrance, as it presents a new approach which is not based on gradient bounds. Combining the theory of Dirichlet forms with methods from numerical analysis we find abstract criteria for Mosco convergence of standard gradient forms with varying reference measures. These include cases in which the measures are not log-concave. To demonstrate the accessibility of our abstract theory we discuss a first application, generalizing an approximation result from [7], which first appeared in 2011.