Probabilistic representation for solutions to nonlinear Fokker-Planck equations

TL;DR

This paper develops a probabilistic framework for representing solutions to nonlinear Fokker-Planck equations, linking them to stochastic differential equations, and extends the approach to equations with linear space-dependent drift.

Contribution

It introduces a probabilistic representation for entropic solutions to nonlinear Fokker-Planck equations, including cases with multivalued nonlinear diffusion and linear space-dependent drift.

Findings

Probabilistic representation for nonlinear Fokker-Planck solutions

Connection to solutions of nonlinear stochastic differential equations

Extension to equations with linear space-dependent drift

Abstract

One obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker-Planck equation in $\mathbb R^d$ with multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear stochastic differential equation. The case of a nonlinear Fokker-Planck equation with linear space dependent drift is also studied.