熵插值的长时间行为
Long-time behaviour of entropic interpolations
摘要：
本文研究熵插值。这些测度值曲线描述了薛定谔问题[Sch 31]的最优解，薛定谔问题是找到一个独立布朗粒子系统的最可能演化的问题。熵插值在短时间内收敛于最优输运的McCann-Geodesics。在这里，我们专注于长时间的行为，证明特别是熵成本的渐近结果，并建立对热方程，这是梯度流的熵根据奥托演算解释熵插值的收敛。假设Bakry-{\displaystyle {\merry}}mery曲率维数条件，也给出了显式速率。在这方面，我们的工作的主要新颖之处之一是，我们能够控制长时间的行为熵插值假设CD（0，n）条件。
英文摘要：
In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schr{ö}dinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent Brownian particles conditionally to observations. It is well known that in the short time limit entropic interpolations converge to the McCann-geodesics of optimal transport. Here we focus on the long-time behaviour, proving in particular asymptotic results for the entropic cost and establishing the convergence of entropic interpolations towards the heat equation, which is the gradient flow of the entropy according to the Otto calculus interpretation. Explicit rates are also given assuming the Bakry-{É}mery curvature-dimension condition. In this respect, one of the main novelties of our work is that we are able to control the long time behavior of entropic interpolations assuming the CD(0, n) condition only.
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