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Brownian motion with variable drift: 0-1 laws, hitting probabilities and Hausdorff dimension

Published online by Cambridge University Press:  04 July 2012

YUVAL PERES  and
PERLA SOUSI
YUVAL PERES
Affiliation:
Microsoft Research, Redmond, Washington, U.S.A. e-mail: peres@microsoft.com
PERLA SOUSI
Affiliation:
University of Cambridge, Cambridge. e-mail: p.sousi@statslab.cam.ac.uk
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Abstract

By the Cameron–Martin theorem, if a function f is in the Dirichlet space D, then B + f has the same a.s. properties as standard Brownian motion, B. In this paper we examine properties of B + f when fD. We start by establishing a general 0-1 law, which in particular implies that for any fixed f, the Hausdorff dimension of the image and the graph of B + f are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function f is Hölder(1/2), then B + f is intersection equivalent to B. Moreover, B + f has double points a.s. in dimensions d ≤ 3, while in d ≥ 4 it does not. We also give examples of functions which are Hölder with exponent less than 1/2, that yield double points in dimensions greater than 4. Finally, we show that for d ≥ 2, the Hausdorff dimension of the image of B + f is a.s. at least the maximum of 2 and the dimension of the image of f.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 2 , September 2012 , pp. 215 - 234
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

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