this paper. In Section 3 we show how some refinements of the Feynman-Kac formula may be understood in terms of a decomposition of the Brownian path at the time of the last visit to zero before time ` where ` is an exponentially distributed random time independent of the Brownian motion. We also show how D.Williams' decomposition at the maximum of the generic excursion under Ito's measure translates in terms of solutions of a Sturm-Liouville equation. Finally, Section 4 is devoted to explicit computations of the laws of