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∇ϕ|∇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.