Brownian motion with variable drift can be space-filling

TL;DR

This paper constructs a specific function to show that Brownian motion with a variable drift can have a range covering an open set, demonstrating space-filling properties in higher dimensions.

Contribution

It provides a new construction of a drift function for Brownian motion that ensures the process's range is space-filling, answering longstanding questions in stochastic analysis.

Findings

Constructed an $ ext{α}$-Hölder continuous drift function for $d ext{-} $dim Brownian motion.

Proved the range of the drifted process covers an open set for $ ext{α} < 1/d$.

Strengthened previous results on the space-filling nature of Brownian motion with variable drift.

Abstract

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $\alpha < 1/d$ we construct an $\alpha$-H\"{o}lder continuous function $f \colon [0,1] \to \mathbb{R}^d$ so that the range of $B-f$ covers an open set. This strengthens a result of Graversen (1982) and answers a question of Le Gall (1988).