Monge-Kantorovich distance and PDEs

The Monge transfer problem goes back to the 18th century. It consists in minimizing the transport cost of a material from a place to another (and changing the shape). Monge could not solve the problem and the next significant step was achieved 150 years later by Kantorovich who introduced the transport distance between two probability measures as well as the dual problem.

The Monge-Kantorovich distance is not easy to use for Partial Differential Equations and the method of doubling the variables is one of them. It is very intuitive in terms of stochastic processes and this provides us with a method for conservative PDEs as parabolic equations (possibly fractional), homogeneous Boltzman equation, scattering equation or porous medium equation...

Structured equations, as they appear in mathematical biology, is a particular class where the method can be used.

Speaker Biography

Benoît Perthame studied at the École Normale Supérieure, and has been a Professor at the University of Orléans, the École Normale Supérieure and Paris VI. He is a leader in the area of non-linear partial differential equations, and has made important contributions both to the theory of differential equations. He has also played a pioneering role in applying differential equations to problems of modeling in biology and other sciences. He has written several research monographs, as well as close to 300 papers.

Benoît Perthame was an Invited Speaker at the ICM in 1994, and gave a plenary lecture at the ICM in 2014. He has received the Peccot Prize from the Collège de France, and is a member of the French Academiy of Sciences.