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Stochastic optimal transport revisited

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Abstract

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker–Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the Mather theory. The superposition principle is the construction of a semimartingale from the Fokker–Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by Nelson, called Nelson’s problem, and was proved by Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrödinger’s problem that is a typical example of the stochastic optimal transportation problem.

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Correspondence to Toshio Mikami.

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This article is part of the topical collection Viscosity solutions, Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, and Senjo Shimizu.

Partially supported by JSPS KAKENHI Grant Numbers JP26400136 and 19K03548.

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Mikami, T. Stochastic optimal transport revisited. SN Partial Differ. Equ. Appl. 2, 5 (2021). https://doi.org/10.1007/s42985-020-00059-3

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