A note on fast times of Brownian motion with variable drift

TL;DR

This paper investigates how adding a variable drift to Brownian motion affects the Hausdorff dimension of its set of fast times, showing that this dimension cannot be reduced by such modifications.

Contribution

It proves that the Hausdorff dimension of the set of fast times remains unchanged or cannot be decreased when a variable drift is added to Brownian motion.

Findings

Hausdorff dimension of fast times is unaffected by added drift

Adding a function to Brownian motion cannot decrease the dimension of fast times

The dimension remains at least as large as in the standard case

Abstract

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.