P. J. Fitzsimmons and Jim Pitman
May 29, 1998
Mark Kac introduced a method for calculating the distribution of the integral for a function v of a Markov process and a suitable random time T, which yields the Feynman-Kac formula for the moment-generating function of . We review Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of , which may be stated as follows. For any random time T such that the killed process is Markov with substochastic semi-group , any non-negative measurable function v, and any initial distribution , the moment of is where is the Green's operator of the killed process, is the operator of multiplication by v, and is the function that is identically 1.