A non-local singular non-linear Fokker-Planck PDE

TL;DR

This paper studies a complex non-local singular non-linear Fokker-Planck PDE, proving existence, uniqueness, and applying results to stochastic differential equations, ensuring mass conservation and positivity.

Contribution

It introduces new analytical techniques for a non-local singular PDE and connects it to McKean stochastic differential equations, establishing well-posedness and probabilistic properties.

Findings

Proved existence and uniqueness of solutions.

Established mass conservation and positivity.

Applied results to non-local singular McKean SDEs.

Abstract

The focus of this paper is a non-local singular non-linear Fokker-Planck partial differential equation (PDE). The peculiarity of this PDE feature is in its divergence coefficient, which presents a product between a Besov distribution and a non-linearity. The latter involves the convolution between an integrable kernel K and the solution of the PDE, which leads to a non-locality of the first order term in the PDE. We prove existence and uniqueness of a solution to the PDE as well as continuity results on its coefficients. Previous analytical results are then applied to the study of well-posedness in law for a non-local singular McKean stochastic differential equation. As byproduct of that probabilistic representation, we establish mass conservation and positivity preserving for the PDE.