On the finiteness of moments of the exit time of planar Brownian motion from comb domains

Abstract

A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite p-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the p-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite p-th moment for all p ≥ 1/2.

On the Hardy number of comb domains

“…This result is stronger than the corollary of the main theorem of Boudabra and Markowsky in [6], where they approach the problem by studying the moments of the exit time of the Brownian motion. In fact, their main theorem implies that if α n grows at most polynomially in n then the Hardy number is infinite.…”

“…For example, they have been studied in relation with the angular derivative (see [13], [16] and references therein), the harmonic measure [3] and the semigroups of holomorphic functions [4]. Moreover, in [6] Boudabra and Markowsky studied the moments of the exit time of planar Brownian motion from comb domains.…”

Planar Brownian Motion and Complex Analysis

“…The first major work in this direction seems to have been by Burkholder in [31], where it was proved among other things that finiteness of the p-th Hardy norm of Ω is equivalent to finiteness of the p 2 -th moment of T (Ω). Subsequent works on this topic include [26,36,52].…”