论文标题
随机最佳运输重新审视
Stochastic optimal transport revisited
论文作者
论文摘要
我们证明了带有凸成本函数随机最佳运输问题的二元定理，而没有规律性假设，而这些假设通常是在动作积分的半半持续性证明中被假定的。在我们的新方法中，我们证明具有凸成本函数的随机最佳运输问题等于Fokker-Planck方程的一类变异问题，这使我们可以重新审视它们。它是由所谓的叠加原则和马瑟理论的想法完成的。叠加原则是从福克 - 普兰克方程式建造半明星，可以被视为一类所谓的边缘问题，这些问题是从给定边缘分布中构建随机过程的所谓边缘问题。 E. Nelson首先在随机力学中考虑了它，称为Nelson的问题，并由E. Carlen First证明。只要是马尔可夫人，Semimartingale被称为尼尔森过程。在一维情况下，我们还考虑了随机最佳运输问题最小化的Markov特性，而非转换成本具有非凸成本。在证明中，叠加原理和最佳运输问题的最小化功能具有凹成本功能的扮演至关重要的作用。最后，我们证明了Schrodinger问题的半腔和Lipschitz连续性，这是随机最佳运输问题的一个典型例子。
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 E. Nelson, called Nelson's problem, and was proved by E. 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 Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.
