Relative Fatou's theorem for [formula omitted]-harmonic functions in bounded κ-fat open sets

Abstract

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 α-stable processes in a bounded C 1 , 1 open set is true. Here we give a probabilistic proof of relative Fatou's theorem for ( − Δ ) α / 2 -harmonic functions (equivalently for symmetric α-stable processes) in bounded κ-fat open set where α ∈ ( 0 , 2 ) . That is, if u is positive ( − Δ ) α / 2 -harmonic function in a bounded κ-fat open set D and h is singular positive ( − Δ ) α / 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 α-stable process in bounded κ-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 , 1 -open set.