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- W2889711298 abstract "It is well-known (see Dvoretzky, Erd{H o}s and Kakutani [8] and Le Gall [12]) that a planar Brownian motion $(B_t)_{tge 0}$ 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, ${mathcal M}_infty^alpha({rm d} x)$-almost everywhere, there exists a continuous nondecreasing additive functional $({mathfrak L}_t^x)_{tge 0}$, called local times at $x$, such that the support of $ {rm d} {mathfrak L}_t^x$ coincides with the level set ${t: B_t=x}$." @default.
- W2889711298 created "2018-09-27" @default.
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- W2889711298 date "2018-08-30" @default.
- W2889711298 modified "2023-09-23" @default.
- W2889711298 title "Points of infinite multiplicity of planar Brownian motion: measures and local times" @default.
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