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

Abstract

Mark Kac introduced a method for calculating the distribution of the integral $\text{A}{}_{\mathrm{v}}\text{=∫}{}_0{}^{\mathrm{T}}\text{v(X}{}_{\mathrm{t}}\text{)}\text{d}\text{t}$ for a function v of a Markov process $\text{(X}{}_{\mathrm{t}}\text{,}\text{t⩾0)}$ and a suitable random time T, which yields the Feynman–Kac formula for the moment-generating function of A_v. We review Kac’s method, with emphasis on an aspect often overlooked. This is Kac’s formula for moments of A_v, which may be stated as follows. For any random time T such that the killed process $\text{(X}{}_{\mathrm{t}}\text{,}\text{0⩽t<T)}$ is Markov with substochastic semi-group $\text{K}{}_{\mathrm{t}}\text{(x,}\text{d}\text{y)=}\text{P}{}_{\mathrm{x}}\text{(X}{}_{\mathrm{t}}\text{∈}\text{d}\text{y,}\text{T>t)}$, any non-negative measurable function v, and any initial distribution λ, the nth moment of A_v is $\text{P}{}_{\mathrm{\lambda }}\text{A}{}_{\mathrm{v}}{}^{\mathrm{n}}\text{=n!λ(GM}{}_{\mathrm{v}}\text{)}{}^{\mathrm{n}}\text{1}$ where $\text{G=∫}{}_0{}^{\mathrm{\infty }}\text{K}{}_{\mathrm{t}}\text{d}\text{t}$ is the Green’s operator of the killed process, M_v is the operator of multiplication by v, and $\text{1}$ is the function that is identically 1.