Monge-Kantorovich distance and PDEs

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The Monge transfer problem goes back to the end of the 18th century. It consists in minimizing the transport cost of a material from a mass distribution to another. 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 a global 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, structured equations, as they appear in mathematical biology.

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