A probabilistic outlook on optimal mass transportation and fluid motions

23.10.2019, 14:00  –  Campus Golm, Haus 9, Raum 2.22
Institutskolloquium

Christian Léonard (Université Paris Nanterre), Marc Arnaudon (Université de Bordeaux)

14:00 Christian Léonard (Université Paris Nanterre): Optimal mass transportation: a probabilistic approach
15:00 Tea and Coffee Break
15:30 Marc Arnaudon (Université de Bordeaux): The motion of incompressible viscous fluids: a probabilistic approach

Abstracts:

Christian Léonard (Université Paris Nanterre): Optimal mass transportation: a probabilistic approach
In Villani's second textbook about optimal transport, a thought experiment entitled the "lazy gas experiment" is presented. It is a variant of an older thought experiment by Schrödinger (1931) described in his own words as follows: "Imagine that you observe a system of diffusing particles in thermal equilibrium. Admit that at some time t₀ their distribution is more or less uniform and that at a later time t₁>t₀ you find a significant spontaneous deviation from this uniformity. The question is: How does such a deviation occur? What is its most likely way?" Nowadays, this statistical physics problem is addressed in terms of large deviations of empirical measures of Brownian motions. We shall explain how it relates to the theory of optimal transport. Following Schrödinger's program, we shall evoke the dynamical theory of Brownian motion (Einstein, Nelson), relative entropy (Boltzmann, Sanov) and optimal transport (Brenier, McCann).

Marc Arnaudon (Université de Bordeaux): The motion of incompressible viscous fluids: a probabilistic approach
There are various approaches to the study of the motion of an incompressible non viscous fluid. One is by means of Euler equations, requiring the knowledge of initial position and speed. Another more recent one is Arnold's approach, which minimizes some energy functional, viewing the solutions as geodesics in some infinite dimensional manifold of diffeomorphisms. Unfortunately, finding a solution to this problem is hard. This is why Brenier introduced a relaxation of Arnold's problem, in considering probability measures in path spaces. He introduced an energy minimization problem, and was able to prove existence and uniqueness of solutions. When these solutions are sufficiently regular, they coincide with the ones of Arnold's problem. In this talk we will review and then extend Brenier's approaches to viscous fluids.