Quantitative relative entropy estimates for interacting particle systems with common noise

TL;DR

This paper establishes quantitative bounds on the relative entropy for large interacting particle systems with common noise, extending previous results beyond Lipschitz interaction kernels and using a reduction to the idiosyncratic case.

Contribution

It provides explicit entropy estimates for systems with common noise and non-Lipschitz interactions, advancing the understanding of propagation of chaos in such stochastic systems.

Findings

Explicit bounds on relative entropy between conditional Liouville and stochastic Fokker-Planck equations.

Extension of entropy estimates beyond Lipschitz interaction kernels.

Reduction technique to the idiosyncratic setting enabling the use of large number laws.

Abstract

We derive quantitative estimates proving the conditional propagation of chaos for large stochastic systems of interacting particles subject to both idiosyncratic and common noise. We obtain explicit bounds on the relative entropy between the conditional Liouville equation and the stochastic Fokker--Planck equation with an interaction kernel \(k\in L^2(\R^d) \cap L^\infty(\R^d)\), extending far beyond the Lipschitz case. Our method relies on reducing the problem to the idiosyncratic setting, which allows us to utilize the exponential law of large numbers by Jabin and Wang~\cite{JabinWang2018} in a pathwise manner.