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July 2020 Points of infinite multiplicity of planar Brownian motion: Measures and local times
Elie Aïdékon, Yueyun Hu, Zhan Shi
Ann. Probab. 48(4): 1785-1825 (July 2020). DOI: 10.1214/19-AOP1407

Abstract

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 $(B_{t})_{t\ge 0}$ 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 $\{{\mathcal{M}}_{\infty }^{\alpha }\}_{0<\alpha <2}$, that are supported by the set of the points of infinite multiplicity. We prove that for any $\alpha \in (0,2)$, almost surely the Hausdorff dimension of ${\mathcal{M}}_{\infty }^{\alpha }$ equals $2-\alpha $, and ${\mathcal{M}}_{\infty }^{\alpha }$ 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).

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Elie Aïdékon. Yueyun Hu. Zhan Shi. "Points of infinite multiplicity of planar Brownian motion: Measures and local times." Ann. Probab. 48 (4) 1785 - 1825, July 2020. https://doi.org/10.1214/19-AOP1407

Information

Received: 1 September 2018; Revised: 1 September 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224960
MathSciNet: MR4124525
Digital Object Identifier: 10.1214/19-AOP1407

Subjects:
Primary: 60J65

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 4 • July 2020
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