Isolated zeros for Brownian motion with variable drift

TL;DR

This paper investigates the zero sets of Brownian motion with variable drift, showing that certain smooth drifts can create isolated zeros and establishing bounds on the Hausdorff dimension of zero sets.

Contribution

It demonstrates that for alpha<1/2, there exist alpha-Hölder continuous drifts making zeros isolated with positive probability, and bounds the Hausdorff dimension of zero sets for general drifts.

Findings

Existence of alpha-Hölder drifts with isolated zeros for alpha<1/2

Zero set Hausdorff dimension at least 1/2 with positive probability

Upper bound of 1/2 on Hausdorff dimension for 1/2-Hölder or bounded variation functions

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.