Pure Jump-Type Continuous Time Markov Processes with Unbounded Rates

Question

We know that the infinitesimal generator $G$ of any pure jump-type continuous time Markov process (on an lcscH state space, say $X$) with bounded jump rates is given by: \begin{equation} Gf(x)=q(x)\int\big(f(y)-f(x)\big)\rho(x,dy), \end{equation} for every bounded measurable $f:X\to\mathbb{R}$, where $q(\cdot)$ is the jump rate function (we have, for a fixed $c>0$, $|q(x)|<c$ for every $x\in X$) and $\rho(x,\cdot)$ is the jump distribution kernel. We know that, since the jump rate function is uniformly bounded by $c$, the domain of $G$ is the set of all bounded measurable real-valued functions on $X$. Further, the boundedness of jump rates implies that the corresponding transition semigroup $(P_t)_{t\geq0}$ can be written as \begin{equation} P_t=e^{tG}\;\;\;\forall t\geq0. \end{equation} This characterization of the semigroup in terms of the generator can't be so easily done in case the jump rate function is unbounded. Even figuring out the domain of $G$ is not an easy task in that case. Can anyone cite any article / book where one can read further about how to find the domain of $G$ in case of unbounded jump rates? Help will be appreciated.