The exit time of planar Brownian motion and the Phragmen-Lindelof principle

TL;DR

This paper uses probabilistic methods to establish a version of the Phragmen-Lindelof principle for planar domains, linking the exit time moments of Brownian motion to the growth of analytic functions.

Contribution

It introduces a probabilistic approach to the Phragmen-Lindelof principle, connecting exit time moments with function growth and constructing domains with finite exit time moments.

Findings

Finite p-th moment of exit time implies boundedness or rapid growth of analytic functions.

Provides a method to construct domains with finite exit time moments.

Offers a probabilistic proof of Hansen's theorem.

Abstract

In this note, we prove a version of the Phragmen-Lindelof principle 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 infinity along some sequence more rapidly than $e^{|z|^{2p}}$. We also present a method for constructing domains whose exit time has finite p-th moment, thereby giving a general Phragmen-Lindelof principle for spiral-like and star-like domains, and in the process giving a probabilistic proof of a theorem of Hansen. A number of auxiliary results are presented as well.