Harmonic measures for a point may form a square

Hansen W, Netuka I (2010)
Advances in Mathematics 225(4): 1830-1839.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Hansen, WolfhardUniBi; Netuka, Ivan
Abstract / Bemerkung
Let X be a Green domain in R-d, d >= 2, x is an element of X, and let M-x(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures mu(U)(x), U open in X, x is an element of U, which is contained in the set of extreme points of M-x(P(X)), is dense in M-x(P(X)). In this paper, it is shown that M-x(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U-0, U-1, U-2, U-3 of x such that the harmonic measures mu(U0)(x) ,..., mu(U3)(x) are pairwise different and mu(U0)(x) + mu(U2)(x) = mu(U1)(x) + mu(U3)(x). In fact, these measures form a square with respect to a natural L-2-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations). (C) 2010 Elsevier Inc. All rights reserved.
Stichworte
Balayage; Brownian motion; Stable process; Sub-Laplacian; Extremal measure; Representing measure; Harmonic measure; Laplace; Simplex; Heat equation; Riesz potentials; operator; Square
Erscheinungsjahr
2010
Zeitschriftentitel
Advances in Mathematics
Band
225
Ausgabe
4
Seite(n)
1830-1839
ISSN
0001-8708
Page URI
https://pub.uni-bielefeld.de/record/1794023

Zitieren

Hansen W, Netuka I. Harmonic measures for a point may form a square. Advances in Mathematics. 2010;225(4):1830-1839.
Hansen, W., & Netuka, I. (2010). Harmonic measures for a point may form a square. Advances in Mathematics, 225(4), 1830-1839. https://doi.org/10.1016/j.aim.2010.04.017
Hansen, W., and Netuka, I. (2010). Harmonic measures for a point may form a square. Advances in Mathematics 225, 1830-1839.
Hansen, W., & Netuka, I., 2010. Harmonic measures for a point may form a square. Advances in Mathematics, 225(4), p 1830-1839.
W. Hansen and I. Netuka, “Harmonic measures for a point may form a square”, Advances in Mathematics, vol. 225, 2010, pp. 1830-1839.
Hansen, W., Netuka, I.: Harmonic measures for a point may form a square. Advances in Mathematics. 225, 1830-1839 (2010).
Hansen, Wolfhard, and Netuka, Ivan. “Harmonic measures for a point may form a square”. Advances in Mathematics 225.4 (2010): 1830-1839.

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