Reflected symmetric α-stable processes and regional fractional Laplacian

Abstract

In this paper we investigate the reflected symmetric α-stable processes and their generators. We show that the generators are regional fractional Laplacians on the closed region. In the case of 1 ≤ α < 2 their existence requires that ∂U/∂n = 0. on the boundary. Among other things we obtain the integration by parts formula of the regional fractional Laplacian and the semi-martingale decomposition of the reflected symmetric α-stable processes.