POINTS OF INFINITE MULTIPLICITY OF PLANAR BROWNIAN MOTION: MEASURES AND LOCAL TIMES

• 成果类型: Article
• 署名作者: Aidekon, Elie; Hu, Yueyun; Shi, Zhan
• 署名单位: Sorbonne Universite; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1407
• 发表日期: 2020
• 页码: 1785-1825
• 关键词: behavior

摘要

It is well known (see Dvoretzky, Erdos and Kakutani (Bull. Res. Council Israel Sect. F 7F (1958) 175-180) and Le Gall (J. Funct. Anal. 71 (1987) 246-262)) that a planar Brownian motion (B-t)(t >= 0) has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the construction of a family of random measures, denoted by {M-infinity(alpha)}(0<2), that are supported by the set of the points of infinite multiplicity. We prove that for any alpha is an element of (0, 2), almost surely the Hausdorff dimension of M-infinity(alpha) equals 2 - alpha, and M(infinity)(alpha )is supported by the set of thick points defined in Bass, Burdzy and Khoshnevisan (Ann. Probab. 22 (1994) 566-625) as well as by that defined in Dembo, Peres, Rosen and Zeitouni (Acta Math. 186 (2001) 239-270).