Isolated zeros for Brownian motion with variable drift

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 if f is 1/2-Hölder continuous or of bounded variation.

arXiv comment22 pages, 8 figures, added Corollary 1.7 and Remark 2.3, updated references and acknowledgments