= Pseudo-Poisson process
{c}
{title2=$X_t=Z_{N_t}$}
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 $e^{-rt}\sum_{n\geq0}(rt)^nK^n/n!$. Conditioning on $N_t$ proves the latter formula, and differentiation at zero gives the former. Its <resolvent operator> is $(\lambda+r)^{-1}\sum_{n\geq0}[r/(\lambda+r)]^nK^n$ for $\lambda>0$, by integrating the nonnegative scalar Poisson weights. The series converge in <operator norm> on bounded <Borel measurable functions>.
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