The fractional Laplacian with reflections

Motivated by the notion of isotropic $\alpha$-stable Lévy 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.