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This paper concerns harmonic measure on the domains that arise when infinitely many disjoint closed discs are removed from the unit disc. It investigates which configurations of discs are unavoidable for Brownian motion, and obtains refinements of related results of Akeroyd, and of Ortega-Cerd\`{a} and Seip., Comment: revised arguments, results unchanged

This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical semi-infinite lines.) It characterizes, in terms of the spacing between the hyperplanes, those domains for which there exist minimal harmonic functions with a certain exponential growth., Comment: 18 pages