Abstract Formulation of Mean-Field Models and Propagation of Chaos

TL;DR

This paper introduces a novel abstract framework for mean-field systems using optimal transport and semigroup theory, broadening analysis tools beyond traditional SDE-based methods, and proves propagation of chaos with applications to Levy-type models.

Contribution

It develops a unified semigroup-based approach to analyze mean-field models and establishes propagation of chaos, including new results for Levy-type systems.

Findings

Framework applicable to various mean-field models

Proved well-posedness of mean-field evolution equations

Established propagation of chaos for specific particle systems

Abstract

In this work, we formulate an abstract framework to study mean-field systems. In contrast to most approaches in the available literature which primarily rely on the analysis of SDEs, ours is based on optimal transport and semigroup theory. This allows for the inclusion of a wider range of mean-field particle systems within a unified structure. This new approach involves: (1) constructing an abstract framework using semigroups and generators; (2) formulating a corresponding mean-field evolution problem, and proving its well-posedness; (3) demonstrating the propagation of chaos for a class of N-particle systems associated with the mean-field model. Our results are readily applicable to various mean-field models. To demonstrate this, we apply our findings to obtain a new result for Levy-type mean-field systems, which encompass the McKean-Vlasov diffusion.