  We investigate well-posedness for martingale solutions of stochastic
differential equations, under low regularity assumptions on their coefficients,
widely extending some results first obtained by A. Figalli. 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, to address well-posedness for ordinary differential equations
in metric measure spaces: in particular, we employ in a systematic way new
representations and inequalities for commutators between smoothing operators
and diffusion generators.
