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.
Citation
Tonci Antunovic. Krzysztof Burdzy. Yuval Peres. Julia Ruscher. "Isolated Zeros for Brownian Motion with Variable Drift." Electron. J. Probab. 16 1793 - 1814, 2011. https://doi.org/10.1214/EJP.v16-927
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