A law of large numbers for kinetic interacting diffusions

TL;DR

This paper proves that the empirical distribution of a system of interacting kinetic particles converges to a non-linear Fokker-Planck PDE under weak initial conditions, extending the law of large numbers to complex stochastic systems.

Contribution

It establishes a new law of large numbers for kinetic interacting diffusions with minimal initial assumptions, connecting particle systems to PDEs.

Findings

Convergence in probability of empirical measures to the PDE solution

Weak initial data assumptions are sufficient for convergence

Extends classical laws of large numbers to kinetic particle systems

Abstract

We study the convergence of the empirical distribution associated with a system of interacting kinetic particles subject to independent Brownian forcing in a finite horizon setting, using some recent progress on kinetic non-linear partial differential equations. Under general assumptions that require only weak convergence on the initial datum -- without assuming independence or moment conditions -- we prove convergence in probability to the corresponding non-linear Fokker-Planck PDE.