A non-local singular non-linear Fokker-Planck PDE
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.
Keywords
Cite
@article{arxiv.2605.11801,
title = {A non-local singular non-linear Fokker-Planck PDE},
author = {Luca Bondi and Elena Issoglio and Francesco Russo},
journal= {arXiv preprint arXiv:2605.11801},
year = {2026}
}