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

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