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Article Dans Une Revue Annals of Probability Année : 2020

Points of infinite multiplicity of planar Brownian motion: measures and local times

Résumé

It is well-known (see Dvoretzky, Erd{\H o}s and Kakutani [8] and Le Gall [12]) 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 [1] as well as by that defined in Dembo, Peres, Rosen and Zeitouni [5]. Our construction also reveals that with probability one, ${\mathcal M}_{\infty}^\alpha(\d x)$-almost everywhere, there exists a continuous nondecreasing additive functional $({\mathfrak L}_t^x)_{t\ge 0}$, called local times at $x$, such that the support of $ \d {\mathfrak L}_t^x$ coincides with the level set $\{t: B_t=x\}$.
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Dates et versions

hal-01876066 , version 1 (18-09-2018)
hal-01876066 , version 2 (29-04-2021)

Identifiants

Citer

Elie Aïdékon, Yueyun Hu, Zhan Shi. Points of infinite multiplicity of planar Brownian motion: measures and local times. Annals of Probability, 2020, 48 (4), pp.1785-1825. ⟨10.1214/19-AOP1407⟩. ⟨hal-01876066v2⟩
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