Brownian motion with variable drift can be space-filling

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $α< 1/d$ we construct an $α$-Hölder 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).

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