Brownian motion with variable drift can be space-filling

Abstract

For $d\ge 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $\alpha <1/d$ we construct an $\alpha$-Hölder continuous function $f:[0,1]\to {\mathbb{R}}^d$ 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).

arXiv comment15 pages, 3 figures