The It\^o SDEs and Fokker--Planck equations with Osgood and Sobolev coefficients

TL;DR

This paper establishes existence and uniqueness of solutions for degenerated Itô SDEs with mixed Osgood and Sobolev coefficients, and proves the uniqueness of solutions to the related Fokker--Planck equations using probabilistic methods.

Contribution

It introduces new conditions on coefficients that allow for well-posedness of degenerated SDEs with irregular coefficients, extending classical results.

Findings

Existence of generalized stochastic flows under mixed regularity conditions.

Uniqueness of solutions to the Fokker--Planck equation for these SDEs.

Application of probabilistic methods to prove Fokker--Planck solution uniqueness.

Abstract

We study the degenerated It\^o SDE on $\mathbb R^d$ whose drift coefficient only fulfills a mixed Osgood and Sobolev regularity. Under suitable assumptions on the gradient of the diffusion coefficient and on the divergence of the drift coefficient, we prove the existence and uniqueness of generalized stochastic flows associated to such equations. We also prove the uniqueness of solutions to the corresponding Fokker--Planck equation by using the probabilistic method.