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Let $Z=(Z_n{)}_{n\ge 0}$ be a Markov chain with transition kernel $K$, and let $N$ be an independent Poisson process of constant rate $r>0$. A Pseudo-Poisson process is $X_t=Z_{N_t}$. On the Banach space $B_b(E)$ of bounded Borel measurable functions, with the supremum norm, writeIts infinitesimal generator, transition semigroup, and resolvent operator are respectivelyand, for $\lambda >0$,The transition semigroup averages $K^n$ against the Poisson distribution of the number of updates. Since $K$ is a Markov operator, $\Vert K\Vert \le 1$; both series converge in operator norm. Updates with $Z_{n+1}=Z_n$ are allowed and do not count as changes of the observed state. The degenerate rate $r=0$ gives $L=0$, $P_t=I$, and $R_{\lambda }=I/\lambda$.

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.