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

中文翻译：

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抛物线最优输运在有界域上的指数收敛

我们研究了由最优质量传输引起的 Monge-Ampere 型抛物线偏微分方程的第二个边值问题的解的渐近行为。我们的主要结果是该演化方程的解对最优传输问题的平稳解的收敛速度为指数。我们为解决线性化问题的一类特殊函数推导出微分 Harnack 不等式。使用这种 Harnack 不等式和特定于质量传输的某些技术，我们控制抛物线方程解的时间振荡，并获得指数收敛。此外，在证明过程中，我们提出了与 Kim 和 McCann 在最优传输背景下引入的伪黎曼框架的联系，这本身就很有趣。