for a Markov switching model $$\frac{\mathrm{d}X(t)}{\mathrm{d}t}=F(X(t),S(t)),$$ where $F(x,s)$ is $C^1$ given $s$ and $S(t)$ is governed by a discrete-state Markov process with transition rate $\lambda_{ss'}(x)$ from state $s$ to state $s'$, the infinitesimal generator is given by $$\mathcal{L}f(x, s)=F(x,s)\nabla_x f + \sum_{s'}\lambda_{ss'}(x)(f(x,s')-f(x,s)).$$
I find the infinitesimal generator is given typically with an assumption of irreducibility of the discrete-state Markov process $S(t)$. I am not an expert in stochastic process and am wondering whether the infinitesimal generator exists if the $S(t)$ is reducible given some fixed $x$. It would be better if someone can give how to show the infinitesimal generator exists.