Exponential convergence of parabolic optimal transport on bounded domains (Q2657025)

The authors 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. They are able to prove an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. They obtain this important exponential convergence and the control of the oscillation in time of solutions to the parabolic equation by deriving a differential Harnack inequality for a special class of functions that solve the linearized problem and by certain techniques specific to mass transport. Additionally, in the course of the proof, they discover an interesting connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport.