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

Abstract

Fix a pair of smooth source and target densities $ρ$ and $ρ^*$ of equal mass, supported on bounded domains $Ω, Ω^* \subset \mathbb{R}^n$. Also fix a cost function $c_0 \in C^{4,α}(\overlineΩ \times \overline{Ω^*})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $Ω$ and $Ω^*$ are uniformly $c_0$- and $c_0^*$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(Ω,ρ)$ and $(Ω^*,ρ^*)$ 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 $u \in C^2_xC^1_t(\overlineΩ\times [0, \infty))$ of this parabolic problem, and convergence of $u(\cdot,t)$ as $t \to \infty$ to a Kantorovich potential for the optimal transport map between $(Ω,ρ)$ and $(Ω^*,ρ^*)$ with cost function $c$. A noteworthy aspect of our work is that $c$ does \emph{not} necessarily satisfy the weak Ma-Trudinger-Wang condition.