On the most visited sites of planar Brownian motion
نویسندگان
چکیده
منابع مشابه
On the most visited sites of planar Brownian motion
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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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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ژورنال
عنوان ژورنال: Electronic Communications in Probability
سال: 2012
ISSN: 1083-589X
DOI: 10.1214/ecp.v17-1809