Strong solutions to stochastic differential equations with rough coefficients

TL;DR

This paper establishes strong existence and pathwise uniqueness for stochastic differential equations with irregular coefficients, using Sobolev and $L^p$ bounds, applicable in various cases including elliptic, one-dimensional, and kinetic scenarios.

Contribution

It introduces a flexible approach relying on direct estimates and PDE bounds to prove strong solutions for SDEs with rough coefficients, without requiring uniform ellipticity.

Findings

Proves strong solutions exist under Sobolev bounds on coefficients.

Establishes pathwise uniqueness for SDEs with irregular coefficients.

Applicable to elliptic, one-dimensional, and kinetic SDE cases.

Abstract

We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^p$ bounds for the solution of the corresponding Fokker-Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the Langevin (kinetic) case.