On the excursion process of Brownian motion

Abstract:

Let $W_0^ + (t)$ denote the scaled excursion process of Brownian motion, and let $l_0^ + (a), 0 \leqslant a,$ be its local time at a. The joint distribution of $l_0^ + (a), \beta (a),$ and $\gamma (a)$ is obtained, where $\beta (a)$ and $\gamma (a)$ are the last exit time and the first passage time of a by $W_0^{ + } (t)$.