The Annals of Probability

Strong solutions to stochastic differential equations with rough coefficients

Nicolas Champagnat and Pierre-Emmanuel Jabin

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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

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

Received: July 2013
Revised: July 2017
First available in Project Euclid: 12 April 2018

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05] 35K10: Second-order parabolic equations

Stochastic differential equations strong solutions pathwise uniqueness Fokker–Planck equation rough drift rough diffusion matrix degenerate diffusion matrix kinetic stochastic differential equations maximal operator


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.

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