Unavoidable sets and harmonic measures living on small sets

Hansen W, Netuka I (2014)
Proceedings of the London Mathematical Society 109(6): 1601-1629.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Autor*in
Hansen, WolfhardUniBi; Netuka, Ivan
Abstract / Bemerkung
Given a connected open set U not equal theta in R-d, d >= 2, a relatively closed set A in U is called unavoidable in U if Brownian motion, starting in x is an element of U \ A and killed when leaving U, hits A almost surely or, equivalently, if the harmonic measure for x with respect to U \ A has mass 1 on A. First, a new criterion for unavoidable sets is proved, which facilitates the construction of smaller and smaller unavoidable sets in U. Starting with an arbitrary champagne subdomain of U (which is obtained omitting a locally finite union of pairwise disjoint closed balls (B) over bar (z, r(z)), z is an element of Z, satisfying sup(z is an element of Z) r(z)/dist(z, U-c) < 1), a combination of the criterion and the existence of small non-polar compact sets of Cantor type yields a set A on which harmonic measures for U \ A are living and which has Hausdorff dimension d - 2 and, if d = 2, logarithmic Hausdorff dimension 1. This can be done also for Riesz potentials (isotropic alpha-stable processes) on Euclidean space and for censored stable processes on C-1,C-1 open subsets. Finally, in the very general setting of a balayage space (X, W) on which the function 1 is harmonic (which covers not only large classes of second-order partial differential equations, but also non-local situations as, for example, given by Riesz potentials, isotropic unimodal Levy processes, or censored stable processes) a construction of champagne subsets X \ A of X with small unavoidable sets A is given which generalizes (and partially improves) recent constructions in the classical case.
Erscheinungsjahr
2014
Zeitschriftentitel
Proceedings of the London Mathematical Society
Band
109
Ausgabe
6
Seite(n)
1601-1629
ISSN
0024-6115
Page URI
https://pub.uni-bielefeld.de/record/2720341

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Hansen W, Netuka I. Unavoidable sets and harmonic measures living on small sets. Proceedings of the London Mathematical Society. 2014;109(6):1601-1629.
Hansen, W., & Netuka, I. (2014). Unavoidable sets and harmonic measures living on small sets. Proceedings of the London Mathematical Society, 109(6), 1601-1629. doi:10.1112/plms/pdu048
Hansen, W., and Netuka, I. (2014). Unavoidable sets and harmonic measures living on small sets. Proceedings of the London Mathematical Society 109, 1601-1629.
Hansen, W., & Netuka, I., 2014. Unavoidable sets and harmonic measures living on small sets. Proceedings of the London Mathematical Society, 109(6), p 1601-1629.
W. Hansen and I. Netuka, “Unavoidable sets and harmonic measures living on small sets”, Proceedings of the London Mathematical Society, vol. 109, 2014, pp. 1601-1629.
Hansen, W., Netuka, I.: Unavoidable sets and harmonic measures living on small sets. Proceedings of the London Mathematical Society. 109, 1601-1629 (2014).
Hansen, Wolfhard, and Netuka, Ivan. “Unavoidable sets and harmonic measures living on small sets”. Proceedings of the London Mathematical Society 109.6 (2014): 1601-1629.