  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).
