  We study strong existence and pathwise uniqueness for stochastic differential
equations in $\RR^d$ 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.
