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, ${\mathcal{M}}_{\infty }^{\alpha }(\mathrm{d}x)$-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 $\mathrm{d}{\mathfrak{L}}_t^x$ coincides with the level set $\{t:B_t=x\}$.

In this version, we add the assumption ${\alpha }_1+\dots +{\alpha }_p<2$ in Section 7.2, and fix some typos