Abstract
We study strong existence and pathwise uniqueness for stochastic dierential equations in R d with rough coecients,
Highlights
We investigate the well-posedness of the stochastic differential equation (SDE) in Rd, d ≥ 1, (1.1)
The originality of the method we develop here is that it allows to decouple the various questions involved in well-posedness
Instead assuming that such bounds have already been obtained through other means, we show how to use the Crippa–DeLellis estimates to directly prove strong existence and pathwise uniqueness for (1.1) without using any other probabilistic ideas or methods
Summary
We study strong existence and pathwise uniqueness for stochastic differential equations in Rd 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 Lp 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. 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
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