Exponential convergence of parabolic optimal transport on bounded domains

Abedin, Farhan; Kitagawa, Jun

doi:10.2140/apde.2020.13.2183

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Exponential convergence of parabolic optimal transport on bounded domains

Farhan Abedin and Jun Kitagawa

Vol. 13 (2020), No. 7, 2183–2204

DOI: 10.2140/apde.2020.13.2183

Abstract

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge–Ampère 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.

Keywords

parabolic optimal transport, Monge–Kantorovich, exponential convergence, Kim–McCann metric, Li–Yau Harnack inequality

Mathematical Subject Classification 2010

Primary: 35K96, 58J35

Milestones

Received: 2 January 2019

Revised: 21 June 2019

Accepted: 6 September 2019

Published: 10 November 2020

Authors

Farhan Abedin
Department of Mathematics Michigan State University East Lansing, MI United States

Jun Kitagawa
Department of Mathematics Michigan State University East Lansing, MI United States

Vol. 13, No. 7, 2020