Weak regularity of Gauss mass transport

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form , where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge–Kantorovich problem, curvature flows theory, and parabolic non-linear PDE's.