Suppose we have a CTMC with $Q$-matrix given by $$q(x,y)=\begin{cases}-c(x)&\text{if }y=x\\c(x)/2&\text{if }|y-x|=1\\0&\text{otherwise}\end{cases},$$ for $c:\mathbb Z\to(0,\infty)$ some function (note: $0$ is not included in the range, which would make implosion possible). I'm wondering whether this chain can be explosive, and if so, how to prove this claim.
I have doubts since the jump chain $(S_n)_{n\in\mathbb N_0}$ is a symmetric random walk, hence $\mathbb ES_n=0,\mathbb ES_n^2=n$, so $\mathbb E|S_n|\sim\mathcal O(\sqrt n)$, and hence it seems reasonable that the number of revisits to the starting state should also be of order $\sqrt n$, which would disprove explosivity for any choice of $c$. This would also follow, I think, since any state $x$ is revisited infinitely often $\mathbb P_x$-a.s.
However, I believe an explosive construction should be possible. The context: I'm looking for functions $c$ for which the above process is not Feller. I think I have a proof for Fellerness of the non-explosive case (in particular the easy case where $c(x)=1$ for all $x\in\mathbb Z$), and given an explosive CTMC, it seems that there does not exist a càdlàg modification of this CTMC (no left limits, since the jump chain does not converge), which would prove that such a process is not Feller. In this context, I'm willing to use the convention that we're in state $\partial$ for all $t\geq\zeta$. If an explosive construction is impossible or difficult, I'm also interested in a function $c$ such that the above process is not Feller.
Any help is much appreciated.
I managed to find an answer myself. Indeed, as I already noted in the question, the jump chain is recurrent, hence the starting state $x$ is visited infinitely often. Then write $\zeta$ for the (first) explosion time and write $T_n=c(S_{n-1})H_n$, where $S_{n-1}$ is the $(n-1)$'th state the jump chain is in, and $H_n$ denote the holding times before the $n$'th jump. Then $(T_n)_{n\in\mathbb N}$ is i.i.d. standard exponential. Writing $(N_k)_{k\in\mathbb N}$ for the times the jump chain visits state $x$, we have $$c(x)\zeta=c(x)\sum_{n\geq1}H_n\geq c(x)\sum_{m\geq1}H_{N_m+1}=\sum_{m\geq1}T_{N_m+1}=+\infty.$$ Hence we have $\mathbb P_x$-a.s. non-explosion, for each starting state $x\in\mathbb Z$. See also Norris, Theorem 2.7.1.