Fractional diffusion in convex domains and reflected isotropic stable processes

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.

💡 Research Summary

The paper investigates the fractional diffusion limit of a kinetic scattering equation posed in a strongly convex bounded domain 𝔇⊂ℝᵈ (d ≥ 2) with diffusive boundary conditions. The microscopic model consists of a particle whose velocity Vₑ(t) is a compound Poisson process with intensity ε⁻¹ and i.i.d. jumps drawn from a rotationally invariant heavy‑tailed distribution F satisfying F(v) ∼ |v|^{−α−d} for some α∈(0,2). The position is Xₑ(t)=x₀+ε^{1/α−1}∫₀ᵗ Vₑ(s) ds. When Xₑ hits the boundary ∂𝔇, the velocity is refreshed according to a second distribution G, restricted to outward directions.

The authors prove that as ε→0 the law of Xₑ converges to that of a reflected isotropic α‑stable process, but the precise reflection mechanism depends on the tail of G. Two regimes are identified:

1. Continuous reflection (R*ₜ). If G possesses a finite moment of order α/2 (i.e. its tail is lighter than |v|^{−α/2−d}), the limiting process never jumps away from the boundary. When a jump of the underlying stable process would cross ∂𝔇, the trajectory is stopped at the unique intersection point of the attempted line segment with ∂𝔇 and then continues continuously inside the domain. This yields a process that behaves like an α‑stable motion in the interior and reflects continuously at the boundary.

2. Jump reflection (R(β)ₜ). If G has a heavy tail G(v) ∼ |v|^{−β−d} with β∈(0,α/2), the limit process may leave the boundary by a jump. Again the attempted jump is truncated at the intersection with ∂𝔇, but now the particle makes a random jump back into the interior whose length is distributed according to the power‑law density |x|^{−β−d}. The parameter β governs how far the particle jumps: small β produces long excursions, large β short ones, yet the overall scaling remains the same for all β∈(0,α/2).

The construction of both reflected processes is carried out via an Itô synthesis based on excursion theory. The authors start from a stable process reflected in a half‑space (the “reference” domain) and consider its excursions away from the boundary. Each excursion is rotated, translated, and stopped at the appropriate time so that its image fits inside 𝔇. The concatenation of these transformed excursions yields a global path. The crucial technical object is the boundary process, defined as the original excursion process time‑changed by its local time on ∂𝔇. This boundary process solves a stochastic differential equation driven by a Poisson random measure whose intensity encodes the joint law of the undershoot and overshoot of an excursion when it attempts to exit the domain. Proving well‑posedness requires delicate geometric estimates: a contraction‑type inequality for the mapping that sends a point outside 𝔇 to the stopping point on ∂𝔇, and precise bounds on the distribution of the undershoot/overshoot pair. These are obtained using the strong convexity of 𝔇 and careful analysis of the Lévy measure of the underlying stable process.

With the boundary process in hand, the authors establish that the concatenated process is Markov and Feller, compute its infinitesimal generator, and identify the associated reflected fractional heat equations. The generator consists of two parts: (i) the interior non‑local operator (−Δ)^{α/2} acting on functions defined in 𝔇, and (ii) a non‑local boundary operator that, in the continuous‑reflection case, reduces to a Neumann‑type condition (∂ₙu=0), while in the jump‑reflection case it becomes an integral operator with kernel proportional to |x−y|^{−β−d} acting on the boundary values. The resulting PDEs describe the evolution of the time‑marginals of R*ₜ and R(β)ₜ respectively.

The paper also contains a thorough pre‑limit analysis: it studies the kinetic scattering process with diffusive boundary, derives uniform estimates on its excursions, proves tightness, and shows convergence of the associated excursion measures. Sections 9 and 10 handle the convergence of the boundary processes and of the full scattering process, employing Skorokhod topologies and careful control of the local time. The authors verify that the limiting reflected processes indeed arise as the scaling limits of the original kinetic model.

Overall, the work makes several substantial contributions:

• It extends the fractional diffusion limit from whole space or half‑space settings to arbitrary strongly convex domains.
• It introduces a novel reflection mechanism for jump processes that combines truncation at the boundary with either continuous re‑entry or power‑law jumps, a feature not captured by previous Dirichlet‑form or censored‑process constructions.
• It provides a complete probabilistic construction (Itô synthesis, excursion concatenation, boundary SDE) together with analytic characterisation (generator, reflected fractional PDE).
• It bridges stochastic process theory, kinetic equations, and non‑local PDEs, offering a framework that can be adapted to other kinetic models (e.g., fractional Fokker‑Planck) or to more general domains.

The results open avenues for further research on non‑local boundary value problems, on asymmetric reflected stable processes, and on numerical schemes that respect the intricate boundary dynamics uncovered here.