Let (Bt: t ≥ 0) be a planar Brownian motion and define a family of gauge functions φα(s) = log(1/s) -α for α > 0. If α < 1 we show that almost surely there exists a point x in the plane such that H φα ({t ≥ 0: Bt = x}) > 0, but if α > 1 almost surely H φα ({t ≥ 0: Bt = x}) = 0 simultaneously for all x ∈ R 2. This resolves a longstanding open problem posed by S. J. Taylor in 1986.
On the most visited sites of planar Brownian motion / Cammarota, Valentina; P., Moerters. - In: ELECTRONIC COMMUNICATIONS IN PROBABILITY. - ISSN 1083-589X. - ELETTRONICO. - 17:0(2012), pp. 1-9. [10.1214/ecp.v17-1809]
On the most visited sites of planar Brownian motion
CAMMAROTA, VALENTINAMembro del Collaboration Group
;
2012
Abstract
Let (Bt: t ≥ 0) be a planar Brownian motion and define a family of gauge functions φα(s) = log(1/s) -α for α > 0. If α < 1 we show that almost surely there exists a point x in the plane such that H φα ({t ≥ 0: Bt = x}) > 0, but if α > 1 almost surely H φα ({t ≥ 0: Bt = x}) = 0 simultaneously for all x ∈ R 2. This resolves a longstanding open problem posed by S. J. Taylor in 1986.File allegati a questo prodotto
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