arXiv subjects

Julia Ruscher

Publications and source records attributed to Julia Ruscher.

3 recordsLinked to original sources

arXiv2012

A note on fast times of Brownian motion with variable drift

A famous result of Orey and Taylor gives the Hausdorff dimension of the set of fast times, that is the set of points where linear Brownian motion moves faster than according to the law of iterated logarithm. In this paper we examine what happens to the set of fast times if a variable drift is added to linear Brownian motion. In particular, we will show that the Hausdorff dimension of the set of fast times cannot be decreased by adding a function to Brownian motion.

math.PR↗

arXiv2012

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

It was shown by Antunović, Burdzy, Peres, and Ruscher that a Cantor function added to one-dimensional Brownian motion has zeros in the middle $α$-Cantor set, $α\in (0,1)$, with positive probability if and only if $α\neq 1/2$. 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.

math.PR↗

arXiv2011

Isolated zeros for Brownian motion with variable drift

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.

math.PR↗