Fractional diffusion limit in convex domain and reflected isotropic stable processes

Vortrag von Prof. Dr. Loïc Béthencourt

Sprecher eingeladen von: Prof. Dr. Jean Bertoin

Datum: 11.03.26  Zeit: 17.15 - 18.45  Raum: Y27H12

In this talk, I will present some recent results obtained with Nicolas Fournier. We are interested in a simple model describing the (random) motion of a particle in a gaz, namely the kinetic scattering equation. When the particle lives in $\mathbb{R}^d$, and when the equilibrium (for the velocities) is heavy-tailed, it is easy to see that the position process, correctly rescaled, converges weakly to an $\alpha$-stable process. We are interested in the case where the particle is confined into a convex smooth domain, by a certain reflection mechanism that I will describe. We show that, according to the boundary condition, two different types of reflected stable process may arise at the limit. After introducing the model and the results, I will mostly explain how we construct the limiting processes by piecing together their excursions inside the domain. If time permits, I will say a few words about the convergence.