Optimal Transportation Problem as Stochastic Mechanics

The origin of the optimal transportation problem is the following problem that G. Monge [64] proposed at Paris Royal Science Academy in 1781: What is the most convenient way to move a sand pile from one place to another? This is called Monge’s problem. In the 20th century, this problem was generalized by L.V. Kantorovich [45], [46] so that one can consider it in a mathematically easier framework and is called the Monge-Kantorovich problem (MKP for short) nowadays . The MKP required the study of the MongeAmpere equation . Many kinds of researches on the MKP have been done rapidly, e.g., the studies on the applications of the MKP to partial differential equations, limit theorems of the probability theory, log-Sobolev’s inequality for probability measures, economics and image processings, on the MKP over the Riemannian manifold and Wiener space and on the geometry of the infinite dimensional space and the MKP (see [1-3, 19, 20, 25, 27, 29, 31, 32, 36, 37, 43, 50, 52, 70-72, 77, 80, 81] and the references therein). We refer the readers to the above references for a general theory. We call the MKP and its generalization the optimal transportation problem and consider it as the problem of a random mechanics (=stochastic mechanics) determined by the least action principle.