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 $T = ϕ\frac{\nabla ϕ}{|\nabla ϕ|}$, 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 nonlinear PDE's.

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