## The Annals of Probability

### Strong solutions to stochastic differential equations with rough coefficients

#### Abstract

We study strong existence and pathwise uniqueness for stochastic differential equations in $\mathbb{R}^{d}$ 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.

#### Article information

Source
Ann. Probab., Volume 46, Number 3 (2018), 1498-1541.

Dates
Revised: July 2017
First available in Project Euclid: 12 April 2018

https://projecteuclid.org/euclid.aop/1523520023

Digital Object Identifier
doi:10.1214/17-AOP1208

Mathematical Reviews number (MathSciNet)
MR3785594

Zentralblatt MATH identifier
06894780

#### Citation

Champagnat, Nicolas; Jabin, Pierre-Emmanuel. Strong solutions to stochastic differential equations with rough coefficients. Ann. Probab. 46 (2018), no. 3, 1498--1541. doi:10.1214/17-AOP1208. https://projecteuclid.org/euclid.aop/1523520023

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