A perturbative approach to the parabolic optimal transport problem for non-MTW costs

Abstract

Fix a pair of smooth source and target densities $\rho$ and ${\rho }^{\ast }$ of equal mass, supported on bounded domains $\Omega ,{\Omega }^{\ast }\subset {\mathbb{R}}^n$. Also fix a cost function $c_0\in C^{4,\alpha }(\overline{\Omega }\times \overline{{\Omega }^{\ast }})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $\Omega$ and ${\Omega }^{\ast }$ are uniformly $c_0$- and $c_0^{\ast }$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(\Omega ,\rho )$ and $({\Omega }^{\ast },{\rho }^{\ast })$ when the cost function $c$ is a sufficiently small $C^4$ perturbation of $c_0$, and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution $u\in C_x^2{C}_t^1(\overline{\Omega }\times [0,\infty ))$ of this parabolic problem, and convergence of $u(\cdot ,t)$ as $t\to \infty$ to a Kantorovich potential for the optimal transport map between $(\Omega ,\rho )$ and $({\Omega }^{\ast },{\rho }^{\ast })$ with cost function $c$. A noteworthy aspect of our work is that $c$ does \emph{not} necessarily satisfy the weak Ma-Trudinger-Wang condition.

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