On the increments of the principal value of Brownian local time

Abstract

Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{\d s \over W(s)} := \lim_{ε\to0} \int_0^t 1_{(|W(s)|> ε)} {\d s \over W(s)} $ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.

23 pages