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Stochastic optimal transport revisited

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  • Toshio Mikami

    (Tsuda University)

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.

Suggested Citation

  • Toshio Mikami, 2021. "Stochastic optimal transport revisited," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-26, February.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:1:d:10.1007_s42985-020-00059-3
    DOI: 10.1007/s42985-020-00059-3
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    References listed on IDEAS

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    1. Mikami, Toshio, 2004. "Covariance kernel and the central limit theorem in the total variation distance," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 257-268, August.
    2. Mikami, Toshio & Thieullen, Michèle, 2006. "Duality theorem for the stochastic optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1815-1835, December.
    3. Mikami, Toshio, 1998. "Equivalent conditions on the central limit theorem for a sequence of probability measures on R," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 237-242, March.
    4. Nelson, Edward, 1984. "Quantum fluctuations — An introduction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 124(1), pages 509-519.
    5. Rüschendorf, L. & Thomsen, W., 1993. "Note on the Schrödinger equation and I-projections," Statistics & Probability Letters, Elsevier, vol. 17(5), pages 369-375, August.
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