The Annals of Probability

Strong solutions to stochastic differential equations with rough coefficients

Nicolas Champagnat and Pierre-Emmanuel Jabin

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1523520023

Digital Object Identifier
doi:10.1214/17-AOP1208

Mathematical Reviews number (MathSciNet)
MR3785594

Zentralblatt MATH identifier
06894780

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

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

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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