# Exponential Convergence of Parabolic Optimal Transport on Bounded Domains

## TL;DR

This paper proves that solutions to a specific parabolic PDE related to optimal mass transport on bounded domains converge exponentially fast to the stationary solution, using a differential Harnack inequality and geometric insights.

## Contribution

It establishes the exponential convergence rate for solutions of a Monge-Ampère type parabolic PDE in optimal transport, connecting it with pseudo-Riemannian geometry.

## Key findings

- Exponential convergence rate of solutions to the stationary state.
- Development of a differential Harnack inequality for the linearized problem.
- Connection with pseudo-Riemannian framework in optimal transport.

## Abstract

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve the linearized problem. Using this Harnack inequality and certain techniques specific to mass transport, we control the oscillation in time of solutions to the parabolic equation, and obtain exponential convergence. Additionally, in the course of the proof, we present a connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport, which is interesting in its own right.