Harmonic measures for a point may form a square

Hansen W, Netuka I. Harmonic measures for a point may form a square. Advances in Mathematics. 2010;225(4):1830-1839.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