Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators.