Pseudo-Poisson process ($X_t=Z_{N_t}$)

A Markov chain $Z$ with transition kernel $K$, sampled at an independent Poisson process $N$ of rate $r$, is a pseudo-Poisson process. Its infinitesimal generator is $r(K-I)$ and its transition semigroup is $e^{-rt}{\sum }_{n\ge 0}(rt{)}^n{K}^n/n!$. Conditioning on $N_t$ proves the latter formula, and differentiation at zero gives the former. Its resolvent operator is $(\lambda +r{)}^{-1}{\sum }_{n\ge 0}[r/(\lambda +r){]}^n{K}^n$ for $\lambda >0$, by integrating the nonnegative scalar Poisson weights. The series converge in operator norm on bounded Borel measurable functions.