Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T06:49:31.711Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
PERLA SOUSI
Affiliation:
University of Cambridge, Cambridge. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Antunović, T., Burdzy, K., Peres, Y. and Ruscher, J. Isolated zeros for Brownian motion with variable drift (2010). Avalaible at http://arxiv.org/abs/1009.3603.Google Scholar
[2]Antunović, T., Peres, Y. and Vermesi, B. Brownian motion with variable drift can be space-filling, (2010). Available at http://arxiv.org/abs/1003.0228.Google Scholar
[3]Benjamini, I., Pemantle, R. and Peres, Y.Martin capacity for Markov chains. Ann. Probab. 23, no. 3 (1995), 1332–1346.CrossRefGoogle Scholar
[4]Cuzick, J.Some local properties of Gaussian vector fields. Ann. Probab. 6, no. 6 (1978), 984–994CrossRefGoogle Scholar
[5]Evans, S. N.Perturbation of functions by the paths of a Lévy process. Math. Proc. Camb. Phil. Soc. 105, no. 2 (1989), 377–380.CrossRefGoogle Scholar
[6]Le Gall, J.-F.Sur les fonctions polaires pour le mouvement brownien. Séminaire de Probabilités, XXII. Lecture Notes in Math. vol. 1321 (Springer, Berlin, 1988), pp. 186–189.CrossRefGoogle Scholar
[7]Graversen, S. E.Polar functions for Brownian motion. Z. Wahrsch. Verw. Gebiete 61 no. 2 (1982), 261–270.CrossRefGoogle Scholar
[8]Kahane, J.-P.Some Random Series of Functions (Cambridge University Press, 1985).Google Scholar
[9]Khoshnevisan, D.Brownian sheet images and Bessel-Riesz capacity. Trans. Amer. Math. Soc. 7 (1999), 2607–2622.CrossRefGoogle Scholar
[10]Kôno, N.Double Points of a Gaussian Sample Path. Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978), 175–180.CrossRefGoogle Scholar
[11]Matilla, P.Geometry of Sets and Measures in Euclidean Spaces Fractals and Rectifiability (Cambridge University Press, 1995).CrossRefGoogle Scholar
[12]Mörters, P. and Peres, Y.Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 30 (2010).Google Scholar