Cantor set zeros of one-dimensional Brownian motion minus Cantor function

Abstract

It was shown by Antunović, Burdzy, Peres, and Ruscher that a Cantor function added to one-dimensional Brownian motion has zeros in the middle $\alpha$-Cantor set, $\alpha \in (0,1)$, with positive probability if and only if $\alpha \ne 1/2$. We give a refined picture by considering a generalized version of middle 1/2-Cantor sets. By allowing the middle 1/2 intervals to vary in size around the value 1/2 at each iteration step we will see that there is a big class of generalized Cantor functions such that if these are added to one-dimensional Brownian motion, there are no zeros lying in the corresponding Cantor set almost surely.

arXiv comment19 pages, improved Theorem 3.3

References in corpus (1)

• A note on fast times of Brownian motion with variable drift

Cited by in corpus (1)

• A note on fast times of Brownian motion with variable drift