Relative Fatou's Theorem for $(-\Delta)^{\alpha/2}$-harmonic Functions in Bounded $\kappa$-fat Open Set

Abstract

We give a probabilistic proof of relative Fatou's theorem for $(-\Delta)^{\alpha/2}$-harmonic functions (equivalently for symmetric $\alpha$-stable processes) in bounded $\kappa$-fat open set where $\alpha \in (0,2)$. That is, if $u$ is positive $(-\Delta)^{\alpha/2}$-harmonic function in a bounded $\kappa$-fat open set $D$ and $h$ is singular positive $(-\Delta)^{\alpha/2}$-harmonic function in $D$, then non-tangential limits of $u/h$ exist almost everywhere with respect to the Martin-representing measure of $h$. 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 $\kappa$-fat open set $D$ through non-local 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.