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Brownian motion with variable drift can be space filling

Authors: Tonći Antunović, Yuval Peres and Brigitta Vermesi
Journal: Proc. Amer. Math. Soc. 139 (2011), 3359-3373
MSC (2010): Primary 60J65, 26A16, 26A30, 28A80
Published electronically: February 10, 2011
MathSciNet review: 2811290
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Abstract: For $ d \geq 2$ let $ B$ be standard $ d$-dimensional Brownian motion. For any $ \alpha < 1/d$ we construct an $ \alpha$-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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Additional Information

Tonći Antunović
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Yuval Peres
Affiliation: Theory Group, Microsoft Research, Redmond, Washington 98052

Brigitta Vermesi
Affiliation: Department of Mathematics, University of Washington, Box 354322, Seattle, Washington 98195 – and – Institute for Pure and Applied Mathematics, 460 Portola Plaza, Los Angeles, California 90095

Keywords: Brownian motion, space-filling curves, Hölder continuity
Received by editor(s): May 21, 2010
Received by editor(s) in revised form: August 11, 2010
Published electronically: February 10, 2011
Additional Notes: The third author was supported by NSF Supplemental Funding DMS-0439872 to UCLA-IPAM, P.I. R. Caflisch
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society

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