Isolated zeros for Brownian motion with variable drift

Details Isolated zeros for Brownian motion with variable drift Isolated zeros for Brownian motion with variable drift

2010-09-19

arxiv.org/abs/1009.3603v2

Mathematics

Probability

22 pages, 8 figures, added Corollary 1.7 and Remark 2.3, updated references and acknowledgments

Scientific paper

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.

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