×

zbMATH — the first resource for mathematics

Nonintersection exponents for Brownian paths. II: Estimates and applications to a random fractal. (English) Zbl 0719.60085
From the author’s abstract: Let \(X\) and \(Y\) be independent two-dimensional Brownian motions, \(X(0)=(0,0)\), \(Y(0)=(\varepsilon,0)\), and let \(p(\varepsilon) = P(X[0,1]\cap Y[0,1] = \emptyset\), \(q(\varepsilon) = \{Y[0,1]\) does not contain a closed loop around \(0\}\). Asymptotic estimates (when \(\varepsilon\to 0)\) of \(p(\varepsilon\)), \(q(\varepsilon\)), and some related probabilities, are given. Let \(F\) be the boundary of the unbounded connected component of \(\mathbb{R}^2\setminus Z[0,1]\), where \(Z(t)=X(t) - tX(1)\) for \(t\in [0,1]\). Then \(F\) is a closed Jordan arc and the Hausdorff dimension of \(F\) is less or equal to \(3/2-1/(4\pi^2)\).
[For part I see Probab. Theory Relat. Fields 84, No. 3, 393–410 (1990; Zbl 0665.60078).]

MSC:
60J65 Brownian motion
60G17 Sample path properties
PDF BibTeX XML Cite
Full Text: DOI