On the expected exit time of planar Brownian motion from simply connected domains

Abstract

This paper presents some results on the expected exit time of Brownian motion from simply connected domains in C. We indicate a way in which Brownian motion sees the identity function and the Koebe function as the smallest and largest analytic functions, respectively, in the Schlicht class. We also give a sharpening of a result of McConnell's concerning the moments of exit times of Schlicht domains. We then show how a simple formula for expected exit time can be applied in a series of examples. Included in the examples given are the expected exit times from given points of a cardioid and regular m-gon, as well as bounds on the expected exit time of an infinite wedge. We also calculate the expected exit time of an infinite strip, and in the process obtain a probabilistic derivation of Euler's result that ζ(2) = ∞ n=1 1 n 2 = π 2 6 . We conclude by showing how the formula can be applied to some domains which are not simply connected.

A method for deriving hypergeometric and related identities from theH2Hardy norm of conformal maps

“…We conclude that This reduction appears to be nontrivial. maps D conformally to V , sending 0 to 0 (a proof of this can also be found in [15]). We find that The formula for G V was presented in [13, Sec.…”

“…These propositions (and much more) were proved by Burkholder in the elegant paper [5] (see Theorem 3.2 and Remark 3.1 of [5], together with Lemma 1.1 of [4] and/or Lemma 1 of [15]). There, the quantities arose naturally in terms of the exit time of Brownian motion from domains.…”

“…In what follows we will write a = x + iy, and any references to x and y will always refer to the real and imaginary parts of a. Examples 1 through 3 appear in [15], but since the approach was somewhat different in that paper it seems as well to briefly review them. In each case G V (z) will be a nonnegative function satisfying (i )-(iii ) on V , which by Proposition 2 is unique if it exists.…”

“…The conformal map from D to V which fixes 0 is given by (see [15] for a quick proof of this standard fact)…”

On the distribution of planar Brownian motion at stopping times

“…. This is Euler's celebrated Basel sum, which has many other existing proofs, including a different probabilistic proof making use of planar Brownian motion (see [4]). Note also that if we differentiate (2.11) 2m − 2 times and let θ −→ 0 we will be able to obtain the well-known values of ∞ k=1 1 k 2m .…”