Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems

TL;DR

This paper establishes the existence and uniqueness of solutions to nonlocal, nonlinear Fokker-Planck equations with Bernstein function operators, linking these solutions to nonlinear Markov processes and fulfilling McKean's program.

Contribution

It proves the existence and uniqueness of solutions to nonlocal nonlinear Fokker-Planck equations and connects them to nonlinear Markov processes, extending McKean's framework.

Findings

Existence and uniqueness of solutions to nonlocal nonlinear Fokker-Planck equations.

Solutions form a nonlinear Markov process in McKean's sense.

One-dimensional marginal laws solve the nonlocal nonlinear Fokker-Planck equation.

Abstract

This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $\Psi(-\Delta)$, where $\Psi$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs.