Well-posedness and mean-field limit of discontinuous weighted dynamics via the relative entropy method

TL;DR

This paper establishes the well-posedness of a class of discontinuous weighted particle dynamics and derives their mean-field limit using the relative entropy method, under mild regularity conditions.

Contribution

It provides the first rigorous proof of existence, uniqueness, and mean-field limit for discontinuous weighted dynamics with entropy estimates.

Findings

Proved existence and uniqueness of solutions for the limit PDE.

Established entropy inequalities for the Kolmogorov equation.

Derived the mean-field limit using the relative entropy method.

Abstract

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.