## Electronic Journal of Probability

### Isolated Zeros for Brownian Motion with Variable Drift

#### Abstract

It is well known that standard one-dimensional Brownian motion $B(t)$ has no isolated zeros almost surely. We show that for any $\alpha&lt;1/2$ there are alpha-Hölder continuous functions $f$ for which the process $B-f$ has isolated zeros with positive probability. We also prove that for any continuous function $f$, the zero set of $B-f$ has Hausdorff dimension at least $1/2$ with positive probability, and $1/2$ is an upper bound on the Hausdorff dimension if $f$ is $1/2$-Hölder continuous or of bounded variation.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 65, 1793-1814.

Dates
Accepted: 27 September 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820234

Digital Object Identifier
doi:10.1214/EJP.v16-927

Mathematical Reviews number (MathSciNet)
MR2842087

Zentralblatt MATH identifier
1245.60075

Rights

#### Citation

Antunovic, Tonci; Burdzy, Krzysztof; Peres, Yuval; Ruscher, Julia. Isolated Zeros for Brownian Motion with Variable Drift. Electron. J. Probab. 16 (2011), paper no. 65, 1793--1814. doi:10.1214/EJP.v16-927. https://projecteuclid.org/euclid.ejp/1464820234

#### References

• Antunović, Tonći; Peres, Yuval; Vermesi, Brigitta. Brownian motion with variable drift can be space-filling. Preprint, available at arXiv:1003.0228.
• Boufoussi, Brahim; Dozzi, Marco; Guerbaz, Raby. Path properties of a class of locally asymptotically self similar processes. Electron. J. Probab. 13 (2008), no. 29, 898–921.
• Bass, Richard F.; Burdzy, Krzysztof. The supremum of Brownian local times on Hölder curves. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 627–642.
• Bass, Richard F.; Burdzy, Krzysztof. Erratum to: "The supremum of Brownian local times on Hölder curves" [Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 6, 627–642; ]. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 5, 799–800.
• Burdzy, Krzysztof; Chen, Zhen-Qing; Sylvester, John. The heat equation and reflected Brownian motion in time-dependent domains. II. Singularities of solutions. J. Funct. Anal. 204 (2003), no. 1, 1–34.
• Dvoretzky, A.; Erdó, P.; Kakutani, S. Nonincrease everywhere of the Brownian motion process. 1961 Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II pp. 103–116 Univ. California Press, Berkeley, Calif. (24 #A2448)
• Graversen, S. E. "Polar"-functions for Brownian motion. Z. Wahrsch. Verw. Gebiete 61 (1982), no. 2, 261–270.
• Hawkes, John. Trees generated by a simple branching process. J. London Math. Soc. (2) 24 (1981), no. 2, 373–384.
• Kahane, Jean-Pierre. Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. xiv+305 pp. ISBN: 0-521-24966-X; 0-521-45602-9
• Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
• Le Gall, Jean-François. Sur les fonctions polaires pour le mouvement brownien. (French) [On polar functions for Brownian motion] Séminaire de Probabilités, XXII, 186–189, Lecture Notes in Math., 1321, Springer, Berlin, 1988.
• Loud, W. S. Functions with prescribed Lipschitz condition. Proc. Amer. Math. Soc. 2, (1951). 358–360.
• Marx, Imanuel; Piranian, George. Lipschitz functions of continuous functions. Pacific J. Math. 3, (1953). 447–459.
• Mattila, Pertti. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. xii+343 pp. ISBN: 0-521-46576-1; 0-521-65595-1
• Mörters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8
• Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
• Taylor, S. J.; Watson, N. A. A Hausdorff measure classification of polar sets for the heat equation. Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 2, 325–344.