The optimal transport problem consists in moving a probability density onto another one, in such a way that a certain cost is minimised. Set up by G. Monge in 1781, this optimisation problem has led to many works since the 1980’s, for the theory on its own and for its links with geometry, analysis of partial differential equations, probability theory and statistics.
In a first part we will study the basic results in the theory : original and weak formulations in a space of measures, existence of weak solutions, dual formulation in a space of functions due to L. Kantorovich, existence of a strong solution due to Y. Brenier, Eulerian and Lagrangian dynamical formulation due to R. J. McCann, solution of the problem in dimension one or in a discrete setting, topology and geometry of the spaces of measures. These results are based on notions in topology and functional analysis (compactness criteria for measures and functions, function/measure duality arguments, min-max theorems, convexity) which are classical but which will be recalled or detailed when needed.
In a second part we will consider applications, notably to geometric (isoperimetric, Brunn-Minkowski) and functional (Sobolev, logarithmic Sobolev) inequalities, partial differential equations (heat, porous medium, Fokker-Planck, Botzmann) and stochastic processes, the mean field limit of large interacting particle systems , or the concentration of measure phonemenon.
Main references
F. Santambrogio. Optimal Transport for Applied Mathematicians. Birkhauser - Springer, Cham, 2015
C. Villani, Topics in Optimal Transportation. American Mathematical Society, Providence, 2003
C. Villani, Optimal Transport: Old and New. Springer, New York, 2008