A law of large numbers for kinetic interacting diffusions
Carlo Bellingeri, Fabio Coppini
TL;DR
This work establishes a law of large numbers for the empirical measure of a system of interacting kinetic diffusions driven by Brownian forcing, showing convergence to the solution of the nonlinear kinetic Fokker-Planck equation $\partial_t\nu_t+ v\cdot\nabla_x \nu_t = \frac{\sigma^2}{2}\Delta_v \nu_t - \mathrm{div}_v(\nu_t (\Gamma*\nu_t))$ under very weak initial-data assumptions. The authors develop a kinetic mild PDE framework based on a kinetic semigroup with kernel $G(t,\xi,\eta)$, and formulate a stochastic PDE for the empirical measure, enabling a probabilistic proof that does not rely on independence or strong moment conditions. Key technical innovations include the use of anisotropic kinetic Besov/Hilbert spaces to handle product estimates and regularization, a careful Fourier-based analysis of the semigroup, and a GRR-based pathwise control of the stochastic convolution. The main result provides an explicit rate bound for the convergence in expectation and establishes convergence in probability when the initial empirical measure converges, extending classical LLN results to kinetic systems with relaxed assumptions and offering a robust framework for non-IID initial data.
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.