A PERTURBATIVE APPROACH TO THE PARABOLIC OPTIMAL TRANSPORT PROBLEM

Fix a pair of smooth source and target densities \rho and \rho\ast of equal mass, supported on bounded domains S2,S2\ast \subset Rn. Also fix a cost function c0 \in C4,\alpha(S2 \times S2\ast) satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume S2 and S2\ast are uniformly c0- and c\ast0- convex with respect to each other. We consider a parabolic version of the optimal transport problem between (S2, \rho ) and (S2\ast, \rho\ast) when the cost function c is a sufficiently small C4 perturbation of c0, and where the size of the perturbation depends on the given data. Our main result establishes global in-time existence of a solution u \in Cx2Ct1 (S2 \times [0, oc)) of this parabolic problem, and convergence of u(center dot, t) as t oc to a Kantorovich potential for the optimal transport map between (S2, \rho ) and (S2\ast, \rho\ast) with cost function c. This is the first convergence result for the parabolic optimal transport problem when the cost function c fails to satisfy the weak Ma--Trudinger--Wang condition by a quantifiable amount.