Kac's formula, levy's local time and brownian excursion

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple The original purpose of this paper was to find a workable expression for the transform of the Brownian excursion area. We wanted to use two well-known results: Kac's formula for Brownian functionals and Levy's local time decompos- ition. We actually found that these two powerful tools lead also to alternative and sometimes simpler proofs for other results on Brownian excursion prob- abilities: maximum and local time distributions. We were also able to derive a useful general result on Brownian excursion symmetric additive functionals. Finally, we also wanted to understand some curious relations that had been observed by earlier authors: connections between hitting-time densities and maximum distributions. Using simple probabilistic arguments and a formula for Jacobi's third 0 function, we can explain these connections in direct terms. The paper is organized as follows. In Section 1, we summarize basic notations and known results. Sections 2 and 3 are short surveys of Kac's and Levy's results. These tools are applied in Sections 4 and 5 to Brownian excursion maximum and local time. Section 6 contains the main results of the paper: a useful new form for the transform of the Brownian excursion area density and a simple relation for functionals based on a symmetric positive function. Section 7 explains those curious relations alluded to earlier between Brownian excursion probabilities. 1. Basic notations and known results