About Fokker-Planck equation with measurable coefficients and applications to the fast diffusion equation

TL;DR

This paper establishes the uniqueness of solutions for a class of Fokker-Planck equations with measurable, possibly degenerate coefficients, and applies these results to probabilistically represent solutions of the fast diffusion equation.

Contribution

It provides new uniqueness results for Fokker-Planck equations with non-smooth coefficients and links these to the probabilistic representation of the Barenblatt solution of the fast diffusion equation.

Findings

Proved uniqueness for Fokker-Planck equations with measurable coefficients.

Established a probabilistic representation for the Barenblatt solution.

Derived uniform small-time density estimates for solutions.

Abstract

The object of this paper is the uniqueness for a $d$-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so called Barenblatt solution of the fast diffusion equation which is the partial differential equation $\partial_t u = \partial^2_{xx} u^m$ with $m\in(0,1)$. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition