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A subharmonicity property of harmonic measures


Author: Vilmos Totik
Journal: Proc. Amer. Math. Soc. 144 (2016), 2073-2079
MSC (2010): Primary 31C12, 31A15, 60J45
DOI: https://doi.org/10.1090/proc/12855
Published electronically: October 1, 2015
MathSciNet review: 3460168
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Abstract: Recently it has been established that for compact sets $ F$ lying on a circle $ S$, the harmonic measure in the complement of $ F$ with respect to any point $ a\in S\setminus F$ has convex density on any arc of $ F$. In this note we give an alternative proof of this fact which is based on random walks, and which also yields an analogue in higher dimensions: for compact sets $ F$ lying on a sphere $ S$ in $ \mathbf {R}^n$, the harmonic measure in the complement of $ F$ with respect to any point $ a\in S\setminus F$ is subharmonic in the interior of $ F$.


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Additional Information

Vilmos Totik
Affiliation: MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, Aradi V. tere 1, 6720, Hungary – and – Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave, CMC342, Tampa, Florida 33620-5700
Email: totik@mail.usf.edu

DOI: https://doi.org/10.1090/proc/12855
Received by editor(s): February 16, 2015
Received by editor(s) in revised form: June 1, 2015
Published electronically: October 1, 2015
Additional Notes: This work was supported by NSF DMS-1265375
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society