The Annals of Probability

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

Krzysztof Burdzy and Gregory F. Lawler

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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 $3/2 - 1/(4\pi^2)$.

Article information

Source
Ann. Probab., Volume 18, Number 3 (1990), 981-1009.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990733

Digital Object Identifier
doi:10.1214/aop/1176990733

Mathematical Reviews number (MathSciNet)
MR1062056

Zentralblatt MATH identifier
0719.60085

JSTOR
links.jstor.org

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G17: Sample path properties

Keywords
Brownian motion fractal intersections of Brownian paths critical exponents

Citation

Burdzy, Krzysztof; Lawler, Gregory F. Nonintersection Exponents for Brownian Paths. II. Estimates and Applications to a Random Fractal. Ann. Probab. 18 (1990), no. 3, 981--1009. doi:10.1214/aop/1176990733. https://projecteuclid.org/euclid.aop/1176990733


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See also

  • Part I: Krzysztof Burdzy, Gregory F. Lawler. Nonintersection Exponents for Brownian Paths. I: Existence and an Invariance Prinicple. Probab. Theory Related Fields, vol. 84, no. 3, 393--410.