The fractional Laplacian with reflections

TL;DR

This paper studies the analytical properties of the fractional Laplacian with reflections, focusing on the associated semigroup, its generator, and convergence to equilibrium within bounded Lipschitz domains.

Contribution

It constructs the transition semigroup, identifies its generator, and proves exponential convergence to a stationary distribution for the reflected fractional Laplacian.

Findings

Construction of the transition semigroup for reflected fractional Laplacian

Identification of the generator of the semigroup

Proof of exponential convergence to stationary distribution

Abstract

Motivated by the notion of isotropic $\alpha$-stable L\'evy processes confined, by reflections, to a bounded open Lipschitz set $D\subset \mathbb{R}^d$, we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time.