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

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.