Electronic Journal of Probability

Isolated Zeros for Brownian Motion with Variable Drift

Tonci Antunovic, Krzysztof Burdzy, Yuval Peres, and Julia Ruscher

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It is well known that standard one-dimensional Brownian motion $B(t)$ has no isolated zeros almost surely. We show that for any $\alpha<1/2$ 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 26A16: Lipschitz (Hölder) classes 26A30: Singular functions, Cantor functions, functions with other special properties 28A78: Hausdorff and packing measures

Brownian motion Hölder continuity Cantor function isolated zeros Hausdorff dimension

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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

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