Regularity properties of the Schrödinger cost

Clerc, Gauthier

doi:10.1051/cocv/2022033

ESAIM: COCV

2022

DOI: 10.1051/cocv/2022033

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Regularity properties of the Schrödinger cost

Gauthier Clerc

Abstract: The Schrödinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.

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Long-time Behaviour of Entropic Interpolations

2022

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In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schrödinger 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-Émery 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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Long-time Behaviour of Entropic Interpolations

2022

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Regularity properties of the Schrödinger cost

Cited by 1 publication

References 22 publications

Long-time Behaviour of Entropic Interpolations

Long-time Behaviour of Entropic Interpolations

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