# Higher order regularity of nonlinear Fokker-Planck PDEs with respect to the measure component

## TL;DR

This paper derives a general formula for higher order derivatives of functionals on the Wasserstein space, applied to solutions of nonlinear Fokker-Planck PDEs, with implications for mean-field games and propagation of chaos.

## Contribution

It introduces a new formula for higher order derivatives of functionals composed with Fokker-Planck PDE solutions, advancing the understanding of measure-dependent PDEs.

## Key findings

- Derived a general formula for higher order linear functional derivatives.
- Established connections with propagation of chaos and mean-field game theory.
- Provides tools for analyzing regularity of nonlinear measure-valued PDEs.

## Abstract

In this article, we establish a general formula for higher order linear functional derivatives for the composition of an arbitrary smooth functional on the 1-Wasserstein space with the solution of a Fokker-Planck PDE. This formula has important links with the theory of propagation of chaos and mean-field games.