Additive Kernels and Integral Representation of Potentials

Abstract

Let P=(P _t ) _t>0 be a submarkovian semigroup of kernels on a measurable space (X,ℬ). An additive kernel of P is a kernel K from X into ]0,∞[ such that P _t K(x,A)=K(x,A+t) for every t>0,x∋X and every Borel subset A of ]0,∞[. It is proved in this paper that for every potential f of P, there exits an additive kernel K of P, unique (up to equivalence) such that f=K1=∫₀ ^∞ K(⋅,dt). This result is already well known for regular potentials of right processes. If U=(U _p ) _p>0 is a sub-Markovian resolvent of kernels on (X,ℬ), we give a notion of additive kernel of U and we prove a similar integral representation of potentials of U.