Fatou's Theorem for censored stable processes

We give a proof of Fatou's Theorem for censored alpha-stable processes in a bounded C-l,C-l open set D where alpha is an element of (1, 2). As an application of Fatou's Theorem, we show that the harmonic measure for such censored alpha-stable process is mutually absolutely continuous with respect to the surface measure of deltaD. Fatou's Theorem is also established for operators obtained from the generator of the censored alpha-stable process through non-local Feynman-Kac transforms. Fatou's Theorem for censored relativistic stable processes is also true as a consequence. (C) 2003 Elsevier B.V. All rights reserved.