On Random Combinatorial Structures and the Local Time of some Brownian Functionals

We present two methods to derive an integral representation of the multi-dimensional densities of the local times of Brownian excursion and reflected Brownian bridge. One is a direct method based on Kac's formula for Brownian functionals, the other is an indirect one using the fact that functionals for Galton-Watson trees resp. random mappings yield corresponding Brownian functionals (excursion resp. bridge) as a limit. This approach is based on generating functions and singularity analysis.