On the Local Time of the Brownian Bridge

Abstract

Let {η(t), 0 ≤ t} ≤ 1} be a standard Brownian bridge. We have for 0 < t < 1, where $$ \Phi (x) = \frac{1}{{\sqrt {{2\pi }} }}\int_{{ - \infty }}^{x} {{{e}^{{ - u \frac{2}{2} }}}} du $$ (4.1) is the normal distribution function. We define $$ \tau (\alpha ) = \mathop{{\lim }}\limits_{{\varepsilon \to 0}} \frac{1}{\varepsilon }{\text{measure\{ }}t:\alpha \leqslant \eta (t) < \alpha + \varepsilon ,0 \leqslant t \leqslant 1{\text{\} }} $$ (4.2) for any real α. The limit (4.2) exists with probability one, and τ(α) is a nonnegative random variable that is called the local time at level α. We have $$ P\{ \tau (\alpha ) \leqslant x\} = 1 - {{e}^{{ - (2|\alpha | + x) \frac{2}{2} }}} $$ (4.3) for x ≥ 0. The notion of local time was introduced by P. Levy [9,10] in 1939 (see also [15] and [6]).