On the Increments of the Principal Value of Brownian Local Time

Abstract

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