Nonintersection Exponents for Brownian Paths. II. Estimates and Applications to a Random Fractal

Abstract

Let $X$ and $Y$ be independent two-dimensional Brownian motions, $X(0) = (0, 0), Y(0) = (\varepsilon, 0)$, and let $p(\varepsilon) = P(X\lbrack 0, 1 \rbrack \cap Y\lbrack 0, 1 \rbrack = \varnothing), q(\varepsilon) = \{Y\lbrack 0, 1 \rbrack \text{does not contain a closed loop around} 0\}$. Asymptotic estimates (when $\varepsilon \rightarrow 0$) of $p(\varepsilon), q(\varepsilon)$, and some related probabilities, are given. Let $F$ be the boundary of the unbounded connected component of $\mathbb{R}^2\backslash Z\lbrack 0, 1 \rbrack$, where $Z(t) = X(t) - tX(1)$ for $t \in \lbrack 0, 1 \rbrack$. Then $F$ is a closed Jordan arc and the Hausdorff dimension of $F$ is less or equal to $3/2 - 1/(4\pi^2)$.