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

Abstract : 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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https://hal.archives-ouvertes.fr/hal-01876066
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Submitted on : Tuesday, September 18, 2018 - 10:32:13 AM
Last modification on : Tuesday, May 5, 2020 - 1:03:21 PM
Document(s) archivé(s) le : Wednesday, December 19, 2018 - 1:41:12 PM

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  • HAL Id : hal-01876066, version 1
  • ARXIV : 1809.07094

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Elie Aïdékon, Yueyun Hu, Zhan Shi. Points of infinite multiplicity of planar Brownian motion: measures and local times. 2018. ⟨hal-01876066⟩

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