Relative Fatou’s Theorem for (−∆)-harmonic Functions in Bounded κ-fat Open Sets

Recently it was shown in Kim [26] that Fatou’s theorem for transient censored α-stable processes in a bounded C open set is true. Here we give a probabilistic proof of relative Fatou’s theorem for (−∆)-harmonic functions (equivalently for symmetric α-stable processes) in bounded κ-fat open set where α ∈ (0, 2). That is, if u is positive (−∆)-harmonic function in a bounded κ-fat open set D and h is singular positive (−∆)-harmonic function in D, then non-tangential limits of u/h exist almost everywhere with respect to the Martin-representing measure of h. This extends the result of Bogdan and Dyda [7]. 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 non-local FeynmanKac transforms. As an application, relative Fatou’s theorem for relativistic stable processes is also true if D is bounded C-open set. AMS 2000 Mathematics Subject Classification: Primary 31B25, 60J75; Secondary 60J45, 60J50