Hansen, WolfhardUniBi; Netuka, Ivan
Abstract / Bemerkung
It is shown that any convex combination of harmonic measures mu(Ul)(x)...,mu(Uk)(x), where U-l,..., U-k are relatively compact open neighborhoods of a given point x is an element of R-d, d >= 2, can be approximated by a sequence (mu(Wn)(x))(n is an element of N) of harmonic measures such that each W-n is an open neighborhood of x in U-l U... U U-k. This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures. This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner. These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hormander's condition, for example, sub-Laplacians on stratified Lie algebras). (c) 2008 Elsevier Inc. All rights reserved.
balayage; Riesz; potentials; Brownian motion; stable process; skorokhod stopping; Harnack's inequalities; green function; capacity density; doubling; property; harmonic space; sub-Laplacian; extremal measure; harmonic measure; Jensen measure
Advances in Mathematics
Hansen W, Netuka I. Convexity properties of harmonic measures. Advances in Mathematics. 2008;218(4):1181-1223.
Hansen, W., & Netuka, I. (2008). Convexity properties of harmonic measures. Advances in Mathematics, 218(4), 1181-1223. doi:10.1016/j.aim.2008.03.009
Hansen, W., and Netuka, I. (2008). Convexity properties of harmonic measures. Advances in Mathematics 218, 1181-1223.
Hansen, W., & Netuka, I., 2008. Convexity properties of harmonic measures. Advances in Mathematics, 218(4), p 1181-1223.
W. Hansen and I. Netuka, “Convexity properties of harmonic measures”, Advances in Mathematics, vol. 218, 2008, pp. 1181-1223.
Hansen, W., Netuka, I.: Convexity properties of harmonic measures. Advances in Mathematics. 218, 1181-1223 (2008).
Hansen, Wolfhard, and Netuka, Ivan. “Convexity properties of harmonic measures”. Advances in Mathematics 218.4 (2008): 1181-1223.