= Solution
Let $Z=(Z_n)_{n\geq0}$ 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>, write
$$
(Kf)(x)=\int_E f(y)K(x,dy),\qquad K^0=I.
$$
Its <infinitesimal generator>, <transition semigroup>, and <resolvent operator> are respectively
$$
\boxed{L=r(K-I),\qquad D(L)=B_b(E),}
$$
$$
\boxed{P_t=e^{tL}=e^{-rt}\sum_{n=0}^{\infty}\frac{(rt)^n}{n!}K^n,}
$$
and, for $\lambda>0$,
$$
\boxed{R_\lambda=\int_0^\infty e^{-\lambda t}P_t\,dt
=\frac1{\lambda+r}\sum_{n=0}^{\infty}\left(\frac r{\lambda+r}\right)^nK^n
=(\lambda I-L)^{-1}.}
$$
The <transition semigroup> averages $K^n$ against the <Poisson distribution> of the number of updates. Since $K$ is a <Markov operator>, $\|K\|\leq1$; 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 $R_\lambda=I/\lambda$.
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