Isolated zeros for Brownian motion with variable drift
Probability
2011-08-17 v2
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-H\"older 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 if f is 1/2-H\"older continuous or of bounded variation.
Keywords
Cite
@article{arxiv.1009.3603,
title = {Isolated zeros for Brownian motion with variable drift},
author = {Tonći Antunović and Krzysztof Burdzy and Yuval Peres and Julia Ruscher},
journal= {arXiv preprint arXiv:1009.3603},
year = {2011}
}
Comments
22 pages, 8 figures, added Corollary 1.7 and Remark 2.3, updated references and acknowledgments