REALIZING STEP FUNCTIONS AS HARMONIC MEASURE DISTRIBUTIONS OF PLANAR DOMAINS

Abstract. The harmonic measure distribution function of a planar domain relates the geometry of the domain to the behaviour of Brow-nian motion in the domain. The value of the function h(r) species the harmonic measure of the part of the boundary of the domain which lies within any given distance r of a xed basepoint in the domain. A longterm goal is to realize all suitable functions as distribution func-tions, by explicit construction of appropriate domains. We show here that increasing step functions can be realized as distribution functions of discs with concentric circular arcs deleted from their interiors. 1