Mosco convergence of gradient forms with non-convex interaction potential

TL;DR

This paper introduces a new approach to establish Mosco convergence of gradient Dirichlet forms with varying measures, extending previous results beyond log-concave measures by combining Dirichlet form theory with numerical analysis methods.

Contribution

It develops abstract criteria for Mosco convergence applicable to non-log-concave measures, broadening the scope of convergence analysis in gradient forms.

Findings

Established criteria for Mosco convergence with non-log-concave measures

Extended convergence results beyond log-concavity assumptions

Demonstrated application to approximation problems in measure spaces

Abstract

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,\mu_N)$ for $N\in\mathbb N$, in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya. The basic assumption is weak measure convergence of the family ${(\mu_N)}_{N}$ on the state space $E$ - either a separable Hilbert space or a locally convex topological vector space. Apart from that, the conditions on ${(\mu_N)}_{N}$ try to impose as little restrictions as possible. The problem has fully been solved if the family ${(\mu_N)}_{N}$ contain only log-concave measures, due to L.~Ambrosio, G.~Savar\'e and L.~Zambotti, 2009. However for a large class of convergence problems the assumption of log-concavity fails. The article suggests a way to overcome this hindrance, as it presents a new approach. 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 by S.~K.~Bounebache and L.~Zambotti, 2014.