The exit time of planar Brownian motion and the Phragmén–Lindelöf principle

Abstract

In this paper a version of the Phragmén-Lindelöf principle is proved using probabilistic techniques. In particular, we will show that if the p th moment of the exit time of Brownian motion from a planar domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to ∞ along some sequence more rapidly than e |z| 2p . We also provide a method of constructing domains whose exit time has finite p th moment. This allows us to give a general Phragmén-Lindelöf principle for spiral-like and star-like domains, as well as a new proof of a theorem of Hansen. A number of auxiliary results are presented as well.

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

“…In fact, coupled with the purely analytic result [12,Thm 4.1] this can be used to determine r@A for any starlike domain in terms of the aperture of at I, which is defined to be the limit as r 3 I of the quantity r; a mxfm@EA X E is a subarc of fjzj a rgg; it is not hard to see that this limit always exists for starlike domains. [25] contains a detailed discussion of this, as well as a version of the Phragmén-Lindelöf principle that makes use of the quantity r@A. Furthermore, the quantity E@@ A p A provides us with an estimate for the tail probability P @ > A: by Markov's inequality, P @ > A E@@A p A p .…”

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

“…In fact, coupled with the purely analytic result [12,Thm 4.1] this can be used to determine H(Ω) for any starlike domain Ω in terms of the aperture of Ω at ∞, which is defined to be the limit as r → ∞ of the quantity α r,Ω = max{m(E) : E is a subarc of Ω ∩ {|z| = r}}; it is not hard to see that this limit always exists for starlike domains. [25] contains a detailed discussion of this, as well as a version of the Phragmén-Lindelöf principle that makes use of the quantity H(Ω). Furthermore, the quantity E((τ Ω ) p ) provides us with an estimate for the tail probability P (τ Ω > δ): by Markov's inequality, P (τ…”

“…x , M − x ⊆ M x . To prove the reverse implication, we apply the following result, which is Theorem 3 in [25].…”

Maximizing the pth moment of the exit time of planar brownian motion from a given domain

“…We would like to conclude that the union of these two strips must then have finite pth moment, but easy examples show that it is not necessarily the case that the union of two domains with finite pth moment must itself have finite pth moment. A method does exist for reaching the desired conclusion, however, and it is contained in Theorem 3 and Lemmas 1 and 2 of [13]. It is straightforward to verify that our infinite strips satisfy the required conditions: their intersection is bounded, and boundary arcs intersect at non-zero angles.…”

“…Therefore the exit time for their union has finite pth moments for all p, and thus so does H . See [13] for details. Now let us see how our methods can be used to localize the pth centers.…”