Uniqueness for a class of stochastic Fokker-Planck and porous media equations

TL;DR

This paper establishes uniqueness results for a broad class of stochastic Fokker-Planck and porous media equations, even with minimal regularity assumptions on coefficients, advancing the theoretical understanding of these stochastic PDEs.

Contribution

It provides the first general uniqueness proofs for stochastic Fokker-Planck and porous media equations with measurable, possibly degenerate coefficients.

Findings

Uniqueness holds under very general assumptions for stochastic Fokker-Planck equations.

Uniqueness is proven for stochastic porous media equations in large function spaces.

The results extend the theoretical framework for stochastic PDEs with irregular coefficients.

Abstract

The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.