Fatouʹs Theorem for censored stable processes

We give a proof of Fatouʹs Theorem for censored α-stable processes in a bounded C1,1 open set D where α∈(1,2). As an application of Fatouʹs Theorem, we show that the harmonic measure for such censored α-stable process is mutually absolutely continuous with respect to the surface measure of ∂D. Fatouʹs Theorem is also established for operators obtained from the generator of the censored α-stable process through non-local Feynman–Kac transforms. Fatouʹs Theorem for censored relativistic stable processes is also true as a consequence.