On the Connected Components of the Complement of a Two-Dimensional Brownian Path

Abstract

Motivated by a question of Mandelbrot [4, Chapter 25], Mountford [5] has investigated the properties of the connected components of the complement of a two-dimensional Brownian path. Let $B = \left( {{B_t},t \geqslant 0} \right)$ be a Brownian motion in the complex plane ℂ, and $B\left[ {0,1} \right] = \left\{ {{B_t},0 \leqslant t \leqslant 1} \right\}$ For any u, y with $0 < \mu < \upsilon \leqslant \infty$, let N [u,υ) denote the number of connected components of ℂ\B[0, 1] with area in [u, υ). Mountford [5] has proved that for every λ ∈ (0, 1),

$${x^2}{\left( {\log x} \right)^2}N\left( {\pi {{\left( {\lambda \chi } \right)}^2},\pi {\chi ^2}} \right)\xrightarrow[{x \to 0}]{{\left( {\Pr obability} \right)}}\frac{1}{2}\left( {{\lambda ^{ - 2}} - 1} \right)$$

(1.a)