|Location:||MSRI: Simons Auditorium|
Kinetic equations involve a large number of variables, time, space and velocity and one important part of the study of those equation consists in giving an approximation of their solution for large time and large observation length. For example, when we model collisions via the linear Boltzmann equation, it is well known that when the equilibria are given by Gaussian distributions, we can approximate the solution by the product of an equilibrium that gives the dependence with respect to velocity multiplied by a density depending on time and position that satisfies a diffusion equation. But different models like inelastic collisions lead to heavy tails equilibria for which depending on the power of the tail, we get different situations. When the diffusion coefficient is no more defined, in the case of linear Boltzmann, the density satisfies a fractional diffusion equation. The same kind of problem arises when the interaction between particles are modeled via the Fokker Planck operator with an additional difficulty. I will present a probabilistic method to study the critical case where we obtain still a diffusion but with an anomalous scaling.
This is a joint work with P. Cattiaux and E.Nasreddine.