Thick points for spatial Brownian motion: Multifractal analysis of occupation measure

• 成果类型: Article
• 署名作者: Dembo, A; Peres, Y; Rosen, J; Zeitouni, O
• 署名单位: Stanford University; City University of New York (CUNY) System; College of Staten Island (CUNY); University of California System; University of California Berkeley; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1-35
• 关键词: formalism; dimension

摘要

Let F(x, r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in R-3. We prove that sup(\x\less than or equal to 1) F(x, r)/(r(2) \log r\) --> 16/pi(2) a.s. as r --> 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a epsilon (0, 16/pi(2)), the Hausdorff dimension of the set of thick points x for which lim sup(r-->0) F(x, r)/(r(2)\log r\) = a is almost surely 2 - a pi(2)/8; this is the correct scaling to obtain a nondegenerate multifractal spectrum for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension d > 3. These results are related to the LIL of Ciesielski and Taylor for the Brownian occupation measure of small balls in the same way that Levy's uniform modulus of continuity, and the formula of Grey and Taylor for the dimension of fast points are related to the usual LIL. We also show that the lim inf scaling of F(x, r) is quite different: we exhibit nonrandom c(1), c(2) > 0, such that c(1) < sup(x) lim inf(r-->0) F(x, r)/r(2) < c(2) a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of limsup type.