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

TL;DR

This paper constructs a family of random measures supported on points of infinite multiplicity of planar Brownian motion, revealing their Hausdorff dimensions and establishing the existence of local times at these points.

Contribution

It introduces a novel family of measures supported on infinite multiplicity points and characterizes their Hausdorff dimensions and local time properties.

Findings

Hausdorff dimension of measures is 2-α for α in (0,2)

Measures supported on thick points of Brownian motion

Existence of local times supported on level sets at these points

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