A law of large numbers for kinetic diffusions

Interacting particle systems are nowadays a well-studied domain with many applications such as modeling biological systems, financial markets and so on. After reviewing some historical results, I will focus on the stochastic setting, where each particle is perturbed by a Brownian motion, and the standard techniques used to prove a Law of Large Numbers. The accent will be posed on the initial condition and the differences and similarities with respect to the deterministic case. While in last setting there is no hypothesis but the convergence of the corresponding empirical measure, in the stochastic setting the current literature generally requires either a strong exchangeability of the particles at time zero, or some technical assumption on the moments. Based on a joint work with Carlo Bellingeri (IECL), I will show how one can recover the deterministic set-up and prove classical Law of Large Numbers for kinetic diffusions. The main tools used are anisotropic Besov spaces, a "simple" SPDE solved by the empirical measure and the Garsia-Rodemich-Rumsey lemma.