NDLI: Boundary Behavior of Harmonic Functions for Truncated Stable Processes

Content Provider	Springer Nature Link
Author	Kim, Panki Song, Renming
Copyright Year	2008
Abstract	For any α∈(0,2), a truncated symmetric α-stable process in ℝ$^{ d }$ is a symmetric Lévy process in ℝ$^{ d }$ with no diffusion part and with a Lévy density given by c\|x\|$^{−d−α }$1$_{{\|x\|<1}}$ for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ$^{ d }$ called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets.
Ending Page	321
Page Count	35
Starting Page	287
File Format	PDF
ISSN	08949840
e-ISSN	15729230
Journal	Journal of Theoretical Probability
Issue Number	2
Volume Number	21
Language	English
Publisher	Springer US
Publisher Date	2008-02-07
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Green functions Relative Fatou type theorem Poisson kernels Truncated symmetric stable processes Probability Theory and Stochastic Processes Probabilistic potential theory Statistics Relative Fatou theorem Harmonic functions Harnack inequality Boundary Harnack principle Jump processes Martin boundary Symmetric stable processes Continuous-time Markov processes on general state spaces Boundary theory
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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