Stochastic optimal transport revisited
Stochastic optimal transport revisited
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DOI：
10.1007/s42985-020-00059-3
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发表时间：
2020-03
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影响因子：
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通讯作者：
T. Mikami
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作者：
T. Mikami
本文证明了具有凸费用函数的随机最优运输问题的对偶定理，而不需要在证明作用积分的下连续性时经常假设的正则性假设。在我们的新方法中，我们证明了具有凸成本函数的随机最优运输问题等价于一类Fokker-Planck方程的变分问题，这让我们重新审视它们。这是通过所谓的叠加原理和马瑟理论的一个想法来实现的。叠加原理是从福克-普朗克方程构造一个半鞅，可以被认为是一类所谓的边际问题，从给定的边际分布构造随机过程。它首先由纳尔逊在随机力学中考虑，称为纳尔逊问题，并首先由卡伦证明。半鞅称为纳尔逊过程，只要它是马尔可夫的。我们还考虑了一维非凸费用的随机最优运输问题的极小值的马尔可夫性质。在证明过程中，叠加原理和凹费用函数最优运输问题的最小值起了关键作用。最后，我们证明了随机最优运输问题的一个典型例子薛定谔问题的可解性和Lipschitz连续性。
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.