PUB - Publikationen an der Universität Bielefeld

The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shotdown process for the fractional Laplacian in Euclidean space. For smooth bounded sets D, we study its transition density and characterize its Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on D. However, for nonconvex D, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for D.