A subharmonicity property of harmonic measures

Totik, Vilmos

doi:10.1090/proc/12855

A subharmonicity property of harmonic measures
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Proc. Amer. Math. Soc. 144 (2016), 2073-2079 Request permission

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 $F$.

References

Kendall Atkinson and Weimin Han, Spherical harmonics and approximations on the unit sphere: an introduction, Lecture Notes in Mathematics, vol. 2044, Springer, Heidelberg, 2012. MR 2934227, DOI 10.1007/978-3-642-25983-8
David Benko, Peter Dragnev, and Vilmos Totik, Convexity of harmonic densities, Rev. Mat. Iberoam. 28 (2012), no. 4, 947–960. MR 2990129, DOI 10.4171/RMI/698
David Benko and Peter D. Dragnev, Balayage ping-pong: a convexity of equilibrium measures, Constr. Approx. 36 (2012), no. 2, 191–214. MR 2957308, DOI 10.1007/s00365-011-9143-x
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original. MR 2450237
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Shizuo Kakutani, Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo 20 (1944), 706–714. MR 14647
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027
Lei Ni, Mean value theorems on manifolds, Asian J. Math. 11 (2007), no. 2, 277–304. MR 2328895, DOI 10.4310/AJM.2007.v11.n2.a6
Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30, Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner. MR 2604525, DOI 10.1017/CBO9780511750489
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
T. J. Willmore, Mean value theorems in harmonic Riemannian spaces, J. London Math. Soc. 25 (1950), 54–57. MR 33408, DOI 10.1112/jlms/s1-25.1.54