Minkowski content of Brownian cut points

TL;DR

This paper proves that the Minkowski content of the set of cut points in Brownian motion in two and three dimensions exists, is finite, and is a non-trivial measure, providing new geometric insights into Brownian paths.

Contribution

It establishes the almost sure existence, finiteness, and non-triviality of the Minkowski content of Brownian cut points in 2D and 3D, a novel geometric characterization.

Findings

Minkowski content of Brownian cut points exists almost surely.

The Minkowski content is finite and non-trivial.

Results apply to Brownian motion in both 2D and 3D.

Abstract

Let $W(t)$, $0\leq t\leq T$, be a Brownian motion in $\mathbb{R}^d$, $d=2,3$. We say that $x$ is a cut point for $W$ if $x=W(t)$ for some $t\in(0,T)$ such that $W [0,t) $ and $W (t,T]$ are disjoint. In this work, we prove that a.s. the Minkowski content of the set of cut points for $W$ exists and is finite and non-trivial.