Non-intersection Exponents for Brownian Paths Part Ii. Estimates and Applications to a Random Fractal
نویسندگان
چکیده
Let X and Y be independent 2-dimensional Brownian motions, X(0) = (0, 0), Y (0) = (ε, 0), and let p(ε) = P (X[0, 1] ∩ Y [0, 1] = ∅), q(ε) = {Y [0, 1] does not contain a closed loop around 0}. Asymptotic estimates (when ε → 0) of p(ε), q(ε), and some related probabilities, are given. Let F be the boundary of the unbounded connected component of R2\Z[0, 1], where Z(t) = X(t) − tX(1) for t ∈ [0, 1]. Then F is a closed Jordan arc and the Hausdorff dimension of F is less or equal to 3/2− 1/(4π2).
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