Relative Fatou's theorem for (-Δ)α/2-harmonic functions in bounded κ-fat open sets

Recently it was shown in [P. Kim, Fatou's theorem for censored stable processes, Stochastic Process. Appl. 108 (1) (2003) 63-92] that Fatou's theorem for transient censored a-stable processes in a bounded C-1,C-1 open set is true. Here we give a probabilistic proof of relative Fatou's theorem for (-Delta)(alpha/2)-harmonic functions (equivalently for symmetric alpha-stable processes) in bounded K-fat open set where a E (0, 2). That is, if it is positive (-Delta)(alpha/2)-harmonic function in a bounded K-fat open set D and It is singular positive (-Delta)(alpha/2)-harmonic function in D, then nontangential limits of u/h exist almost everywhere with respect to the Martin-representing measure of h. This extends the result of Bogdan and Dyda [K. Bogdan, B. Dyda, Relative Fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain, Studia Math. 157 (1) (2003) 83-96]. It is also shown that, under the gaugeability assumption, relative Fatou's theorem is true for operators obtained from the generator of the killed alpha-stable process in bounded K-fat open set D through nonlocal Feynman-Kac transforms. As an application, relative Fatou's theorem for relativistic stable processes is also true if D is bounded C-1,C-1-open set. (c) 2005 Elsevier Inc. All rights reserved.