Nonlinear rough Fokker-Planck equations

TL;DR

This paper introduces rough path techniques to analyze nonlinear stochastic Fokker-Planck equations with common noise, significantly reducing regularity assumptions needed for well-posedness compared to traditional methods.

Contribution

It demonstrates that rough path methods can establish well-posedness of nonlinear stochastic PDEs with less regularity requirements than classical stochastic analysis approaches.

Findings

Rough path techniques effectively handle nonlinear SPDEs with common noise.

Reduced regularity conditions for coefficients compared to previous results.

Enhanced understanding of measure-valued stochastic PDEs in mean-field games.

Abstract

McKean-Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker-Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and conditional law dependence, the evolution becomes random and is governed by a stochastic Fokker-Planck equation; that is, a nonlinear, nonlocal SPDE in the space of measures. (Such equations constitute an important ingredient in the theory of mean-field games with common noise.) Well-posedness of such SPDEs is a difficult problem, the best result to date due to Coghi-Gess (2019), which however comes with dimension-dependent regularity assumptions. In the present work, we show how rough path techniques can circumvent these entirely. Hence, and somewhat contrarily to common believe, the use of rough paths leads to substantially less regularity demands on the coefficients than methods rooted in classical stochastic analysis methods.