# Rough nonlocal diffusions

## TL;DR

This paper develops a new rough path framework to analyze nonlinear Fokker-Planck equations driven by rough signals, specifically addressing McKean-Vlasov diffusions with common noise, and establishes their well-posedness.

## Contribution

It introduces a self-contained nonlinear rough integration theory and defines solutions for the associated Fokker-Planck equations, proving their well-posedness.

## Key findings

- Established a well-posedness theory for rough McKean-Vlasov equations.
- Developed a novel nonlinear rough integration framework.
- Provided a solution concept for rough Fokker-Planck equations.

## Abstract

We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.