On the explicit form of the density of Brownian excursion local time

Abstract:

Let $\underline W _0^ + (t)$, $0 \leqslant t \leqslant 1$, denote Brownian excursion and let $\underline {l} _0^ + (\upsilon )$, $\upsilon \geqslant 0$, be its local time at level $\upsilon$. Starting from a representation of the density of $\underline {l} _0^ + (\upsilon )$ as a complex integral we derive an explicit form of this density, written as an infinite series involving the $n$-fold convolution of known densities. Finally the result is used as an alternative check of Knight’s result on the same topic.