Gradient flow structure for McKean-Vlasov equations on discrete spaces

TL;DR

This paper demonstrates that certain non-linear mean-field equations on discrete spaces can be understood as gradient flows of a free energy functional, with this structure emerging as the limit of N-particle dynamics.

Contribution

It establishes a gradient flow framework for McKean-Vlasov equations on discrete spaces and connects it to particle system limits.

Findings

Gradient flow structure for mean-field equations on discrete spaces

Limit of particle system gradient flows as N approaches infinity

Explicit metric structure for the gradient flow

Abstract

In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity.