How round are the complementary components of planar Brownian motion?

Abstract

Consider a Brownian motion $W$ in $\mathbf{C}$ started from $0$ and run for time 1. Let $A(1),A(2),\dots$ denote the bounded connected components of $\mathbf{C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in \mathbf{N}$. Our main result is that $\mathbf{E}[{\sum }_{i}R(i{)}^2|\log R(i){|}^{\theta }]<\infty$ 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.

28 pages, 11 figures