Pathwise quantitative particle approximation of nonlinear stochastic Fokker-Planck equations via relative entropy

TL;DR

This paper develops a pathwise particle approximation method for nonlinear stochastic Fokker-Planck equations, providing quantitative bounds and establishing existence and uniqueness of solutions using entropy-based techniques.

Contribution

It introduces a novel entropy-based approach to derive and analyze particle approximations for nonlinear stochastic Fokker-Planck equations, applicable to various kernels.

Findings

Derived non-linear stochastic Fokker-Planck equations from particle systems.

Proved existence and uniqueness of solutions using entropy methods.

Provided pathwise quantitative bounds for the particle approximation.

Abstract

We derive non-linear stochastic Fokker-Planck equation from stochastic systems particles with individual and environmental noise via relative entropy method, with pathwise quantitative bounds. Moreover, we prove the existence of a unique strong solution to the associated Fokker-Planck equation. Our proof is based on tools from PDE analysis, stochastic analysis, functional inequalities, and also we use the dissipation of entropy which provides some bound on the Fisher information of the particle system. The approach applies to repulsive and attractive kernels.