Brownian bricklayer: a random space-filling curve

TL;DR

This paper introduces a novel space-filling curve constructed from a Wiener process and its local time, demonstrating a stochastic process that covers the upper half-plane at a constant rate.

Contribution

It presents a new construction of a space-filling curve using Brownian motion and local time, linking stochastic processes with geometric covering properties.

Findings

The curve fills the upper half-plane continuously.

It covers one unit of area per unit time.

The construction connects Brownian motion with geometric measure theory.

Abstract

Let $(B(t),\,t\ge0)$ denote the standard, one-dimensional Wiener process and $(\ell(y,t);\, y\in\mathbb{R},\, t\ge0)$ its local time at level $y$ up to time $t$. Then $\big( (B(t),\, \ell(B(t),t)),\; t\ge0 \big)$ is a random path that fills the upper half-plane, covering one unit of area per unit time.