David Singman

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We study the potential theory of trees with nearest-neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non-negative function H, harmonic outside the root e and(More)
A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many(More)
In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we(More)
In this paper, we give a new definition of the flux of a superharmonic function defined outside a compact set in a Brelot space without positive potentials. We also give a new notion of potential in a BS space (that is, a harmonic space without positive potentials containing the constants) which leads to a Riesz decomposition theorem for the class of(More)
Consider the potentials associated with the Weinstein equation with parameter k in IR ∑∞ j=1 ∂2u ∂xj + k xn ∂u ∂xn = 0, on the upper half space in IRn. We look at translates of curves, Γτ , which lie in the (x1, xn) plane and meet the boundary with degree of tangency τ . We define an α dimensional, non-isotropic Hausdorff measure Hτ α on the boundary, IR(More)
Abstract. Let T be a homogeneous tree of homogeneity q + 1. Let denote the boundary of T, consisting of all infinite geodesics b = [b0, b1, b2, . . .] beginning at the root, 0. For each b ∈ , τ 1, and a 0 we define the approach region τ,a(b) to be the set of all vertices t such that, for some j , t is a descendant of bj and the geodesic distance of t to bj(More)
Abstract. We consider potentials Gkμ associated with the Weinstein equation with parameter k in R, ∑n j=1(∂u/∂x j ) + (k/xn)(∂u/∂xn) = 0, on the upper half space in Rn. We show that if the representing measure μ satisfies the growth condition ∫ yω n /(1 + |y|)n−k < ∞, where max(k, 2− n) < ω 6 1, then Gkμ has a minimal fine limit of 0 at every boundary point(More)