How round are the complementary components of planar Brownian motion?

TL;DR

This paper investigates the geometric properties of the components formed by planar Brownian motion, showing that most components are relatively round, with specific bounds on their radii and shape regularity.

Contribution

It establishes finiteness of expected sums involving out-radius and proves the almost sure divergence of sums involving in-radius, revealing the typical roundness of Brownian motion components.

Findings

Expected sum of squared out-radius times a logarithmic factor is finite.

Sum of squared in-radius times a logarithmic factor diverges almost surely.

Most components exhibit a regular or round shape.

Abstract

Consider a Brownian motion $W$ in ${\bf C}$ started from $0$ and run for time 1. Let $A(1),A(2),\dots$ denote the bounded connected components of ${\bf C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\bf N$. Our main result is that ${\bf E}[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty$ for any $\theta<1$. We also prove that $\sum_i r(i)^2|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.