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Fall and Winter 2016 Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets
Hyunchul Park
Illinois J. Math. 60(3-4): 761-790 (Fall and Winter 2016). DOI: 10.1215/ijm/1506067290

Abstract

We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

Citation

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Hyunchul Park. "Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets." Illinois J. Math. 60 (3-4) 761 - 790, Fall and Winter 2016. https://doi.org/10.1215/ijm/1506067290

Information

Received: 15 September 2016; Revised: 3 April 2017; Published: Fall and Winter 2016
First available in Project Euclid: 22 September 2017

zbMATH: 1376.31003
MathSciNet: MR3705444
Digital Object Identifier: 10.1215/ijm/1506067290

Subjects:
Primary: 31B25, 60J75
Secondary: 60J45, 60J50

Rights: Copyright © 2016 University of Illinois at Urbana-Champaign

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Vol.60 • No. 3-4 • Fall and Winter 2016
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