Minimal positive harmonic functions

Abstract

For the three-dimensional case under hypotheses related to bounded curvature of the boundaries, see C. de la Vall6e Poussin, Propriitis des fonctions harmoniques dans un domaine ouvert limite par des surfaces d courbure bornee,

Resolvents and Martin boundaries of product spaces

“…The reason we use the log blow-up at all faces is that as we shall prove in the next section, the asymptotic behaviour of the resolvent involves expansions in powers of these particular defining functions. The other compactification we consider here is due to Martin [12], and uses the function theory of ∆ g to associate a set of ideal boundary points ∂Z to Z. Notably, it may be carried out in great generality for pairs (Z, H) where H is a semibounded self-adjoint elliptic operator on a space Z, though we shall always assume here that H is the Laplacian.…”

Martin Kernels for Markov Processes with Jumps

“…When D is a Lipschitz domain, then in fact c(D, x 0 , r, R) is of order r β as r → 0 + for some β > 0, which means that f/g extends to a Hölder continuous function at x 0 . A closely related concept of Martin representation of positive harmonic functions was introduced by R. S. Martin in his beautiful article [32], more than three decades before the boundary Harnack inequality became available. Given the existence of limits (2) …”

Book Review: Diffusions, superdiffusions and partial differential equations

“…Given a Greenian domain D, a representation of all positive harmonic functions was obtained by Martin [19] in terms of a compactificationD. Points in the Martin boundary∂D =D\D can be identified with harmonic functions, and an arbitrary positive harmonic function can be represented by the Poisson integral…”