Open Access
Translator Disclaimer
May 2018 Strong solutions to stochastic differential equations with rough coefficients
Nicolas Champagnat, Pierre-Emmanuel Jabin
Ann. Probab. 46(3): 1498-1541 (May 2018). DOI: 10.1214/17-AOP1208


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.


Download Citation

Nicolas Champagnat. Pierre-Emmanuel Jabin. "Strong solutions to stochastic differential equations with rough coefficients." Ann. Probab. 46 (3) 1498 - 1541, May 2018.


Received: 1 July 2013; Revised: 1 July 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894780
MathSciNet: MR3785594
Digital Object Identifier: 10.1214/17-AOP1208

Primary: 35K10 , 60H10 , 60J60

Keywords: degenerate diffusion matrix , Fokker–Planck equation , kinetic stochastic differential equations , Maximal operator , Pathwise uniqueness , rough diffusion matrix , rough drift , Stochastic differential equations , strong solutions

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • May 2018
Back to Top