Isolated Zeros for Brownian Motion with Variable Drift

Tonci Antunovic (University of California Berkeley)
Krzysztof Burdzy (University of Washington)
Yuval Peres (Microsoft Research)
Julia Ruscher (Technische Universität Berlin)

Abstract


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

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Pages: 1793-1814

Publication Date: September 27, 2011

DOI: 10.1214/EJP.v16-927

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