How round are the complementary components of planar Brownian motion?
نویسندگان
چکیده
منابع مشابه
Logarithmic Potentials and Planar Brownian Motion
In Section 5, we saw that for a Brownian motion process in n _ 3 dimensions, P (limtxIX, = o0) = 1 for all x. In sharp contrast to this situation, a planar Brownian motion is certain to hit any nonpolar set. THEOREM 8.1. Let B be a Borel set. Then PX(VB < cX) is either identically 1 or identically 0. PROOF. A simple computation shows that for any x e R2, 1' p(s, x) ds T co as t T oc. Thus, for ...
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Let (Bt : t ≥ 0) be a standard planar Brownian motion. Dvoretzky, Erdős and Kakutani (1958) first showed that, almost surely, there exist points x in the plane such that {t ≥ 0: Bt = x}, the set of times where the Brownian path visits x, is uncountably infinite. Modern proofs of this fact are given in Le Gall (1987) and Mörters and Peres (2010). The result naturally raises the question: How lar...
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ژورنال
عنوان ژورنال: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
سال: 2019
ISSN: 0246-0203
DOI: 10.1214/18-aihp902