Harmonic measures for a point may form a square |
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Authors: | Wolfhard Hansen Ivan Netuka |
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Affiliation: | a Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany b Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75 Praha 8, Czech Republic |
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Abstract: | Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U0, U1, U2, U3 of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L2-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). |
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Keywords: | 30C85 31A15 31D05 35K05 35R03 46A55 58J35 60G52 |
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