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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
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