Fractional diffusion in convex domains and reflected isotropic stable processes

Abstract

We establish the fractional diffusion limit of the kinetic scattering equation with diffusive boundary condition in a strongly convex bounded domain $\mathcal{D}\subset\mathbb{R}^d$. According to the nature of the boundary condition, two types of fractional heat equations may arise at the limit, corresponding to two types of isotropic stable processes reflected in $\mathcal{D}$. In both cases, when the process tries to jump across the boundary, it is stopped at the unique point where $\partial\mathcal{D}$ intersects the line segment defined by the attempted jump. It then leaves the boundary either continuously (for the first type) or by a power-law distributed jump (for the second type). The construction of these processes is done via an It\^o synthesis: we concatenate their excursions in the domain, which are obtained by translating, rotating and stopping the excursions of some stable processes reflected in the half-space. The key ingredient in this procedure is the construction of the boundary processes, i.e. the processes time-changed by their local time on the boundary, which solve stochastic differential equations driven by some Poisson measures of excursions. The well-posedness of these boundary processes relies on delicate estimates involving some geometric inequalities and the laws of the undershoot and overshoot of the excursion when it leaves the domain. We show that these reflected Markov processes are Markov and Feller, we study their infinitesimal generator and we write down the reflected fractional heat equations satisfied by their time-marginals.