Brownian Motion With Variable Drift Can Be Space-Filling

For d ≥ 2 let B be standard d-dimensional Brownian motion. For any α < 1/d we construct an α-Hölder continuous function f : [0, 1] → 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).