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- W1641290693 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." @default.
- W1641290693 created "2016-06-24" @default.
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- W1641290693 date "2004-01-23" @default.
- W1641290693 modified "2023-09-27" @default.
- W1641290693 title "Relative Fatou's Theorem for $(-Delta)^{alpha/2}$-harmonic Functions in Bounded $kappa$-fat Open Set" @default.
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