On the excursion process of Brownian motion

Abstract

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