Fractional diffusion in convex domains and reflected isotropic stable processes
Loïc Béthencourt, Nicolas Fournier
TL;DR
This work analyzes fractional diffusion limits for kinetic scattering in strongly convex domains with diffusive boundary conditions, showing that the position converges to a reflected isotropic α-stable process. It introduces an Itô-style excursion synthesis to construct two classes of reflected processes, governed by boundary interactions: one with continuous exit and one with power-law jumps controlled by β∈(0,α/2). The authors establish well-posedness, Markov/Feller properties, and generators for these processes, and prove convergence of the scaled kinetic scattering model to the corresponding reflected processes under broad tail conditions on the boundary distribution G. The results extend prior half-space analyses to general convex domains and reveal a subcritical vs. supercritical regime depending on moment conditions of G, with a unified approach via excursion measures and boundary-time changes. The methods provide a novel framework for constructing and analyzing reflected jump Markov processes in nontrivial geometries and connect probabilistic excursion theory to fractional PDEs with nonlocal boundary conditions.
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 Itô 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.