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

Abstract

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