The Martin Kernel for Unbounded Domains

The unbounded domains under consideration include the parabolic-type regions and shrinking one-sided horns that have appeared in the literature. We show the Martin boundary at ∞ is a singleton and we determine the growth of the corresponding Martin kernel. For a parabolic-type region expanding at a power law rate, the growth is subexponential and for a horn shrinking at a power law rate it is superexponential.