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A comb domain is defined to be the entire complex plain with a collection ofvertical slits, symmetric over the real axis, removed. In this paper, weconsider the question of determining whether the exit time of planar Brownianmotion from such a domain has finite $p$-th moment. This question has beenaddressed before in relation to starlike domains, but these previous results donot apply to comb domains. Our main result is a sufficient condition on thelocation of the slits which ensures that the $p$-th moment of the exit time isfinite. Several auxiliary results are also presented, including a constructionof a comb domain whose exit time has infinite $p$-th moment for all $p \geq1/2$.

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