Complex analytic proofs of two probabilistic theorems

TL;DR

The paper develops purely analytic proofs of two probabilistic results in complex analysis, replacing Brownian-motion arguments with Hardy-space and conformal-mapping techniques. It establishes a Phragmén-Lindelöf-type principle on simply connected domains under finiteness of the Hardy-p norm ${\cal H}_{p}(W)$, and derives a new analytic expression for the Green's function of the disk that yields Euler-type infinite product identities for $\sinh$, $\cosh$, $\sin$, and $\cos$. The approach includes a detailed treatment of spiral-like and star-like domains, and demonstrates how Green's functions can be understood via covering maps without probabilistic tools. These results unify analytic and probabilistic viewpoints and produce classical product expansions as direct analytic consequences.

Abstract

In this paper, we use purely complex analytic techniques to prove two results of the first author which were hitherto given only probabilistic proofs. A general form of the Phragmén-Lindelöf principle states that if the $p$\textsuperscript{th} Hardy norm of the conformal map from the disk to a simply connected domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to $\ff$ along some sequence more rapidly than $e^{|z|^{p}}$. We will prove this and discuss a number of special cases. We also derive a series expansion for the Green's function of a disk, and show how it leads to an infinite product identity. The celebrated infinite product expansions for sine and cosine are realized as special cases.

Theorems & Definitions (10)

• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Corollary 3
• Corollary 4
• Lemma 1
• Definition 1
• Theorem 3
• proof : Proof of Theorem 3