Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients

TL;DR

This paper establishes the well-posedness of multidimensional diffusion processes with weakly differentiable coefficients, extending prior results by connecting various descriptions and proving existence and uniqueness under minimal regularity.

Contribution

It provides a general equivalence between different formulations of diffusion processes and introduces new existence and uniqueness results for weakly differentiable coefficients.

Findings

Proves equivalence between Fokker-Planck equations and martingale problems.

Establishes existence and uniqueness of solutions with low regularity coefficients.

Employs novel commutator inequalities and energy estimates.

Abstract

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.