Kac’s moment formula and the Feynman–Kac formula for additive functionals of a Markov process

Mark Kac introduced a method for calculating the distribution of the integral Av=∫0Tv(Xt) dt for a function v of a Markov process (Xt, t⩾0) and a suitable random time T, which yields the Feynman–Kac formula for the moment-generating function of Av. We review Kac’s method, with emphasis on an aspect often overlooked. This is Kac’s formula for moments of Av, which may be stated as follows. For any random time T such that the killed process (Xt, 0⩽t<T) is Markov with substochastic semi-group Kt(x,dy)=Px (Xt∈dy, T>t), any non-negative measurable function v, and any initial distribution λ, the nth moment of Av is PλAvn=n!λ(GMv)n1 where G=∫0∞Kt dt is the Green’s operator of the killed process, Mv is the operator of multiplication by v, and 1 is the function that is identically 1.