Variational approach to coarse-graining of generalized gradient flows

TL;DR

This paper introduces a variational method based on duality for coarse-graining gradient flows, effectively handling limits and non-dissipative effects, demonstrated on kinetic equations and Hamiltonian systems.

Contribution

It develops a unified variational approach leveraging duality for coarse-graining in gradient flows, applicable to systems with non-dissipative effects and approximate solutions.

Findings

Successfully applied to overdamped Vlasov-Fokker-Planck limit

Handled small-noise limits in Hamiltonian systems

Extended to approximate solutions with errors

Abstract

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (A) a natural interaction between the duality structure and the coarse-graining, (B) application to systems with non-dissipative effects, and (C) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.