Let $G_1$ and $G_2$ be generators for two distinct continuous-time Markov processes $X^{(1)}$ and $X^{(2)}$ on a common probability space $\Omega$ (with Markov semigroups $S^{(1)}$ and $S^{(2)}$) so that $X^{(1)}_t$ and $X^{(2)}_t$ have states in $\mathbf X,$ i.e. $X^{(i)} : [0,\infty)\times \Omega\rightarrow \mathbf X$.
When is the linear combination $G=aG_1+bG_2$ a Markov generator? (With semigroup $S$.) Intuitively, I feel like the answer should be yes, at least in some cases, but worry there are some technical details I need to address. E.g. maybe there are some assumptions I need to include for the two processes. Maybe $a,b>0$ always works, but if either $a$ or $b$ is negative, they can't be too negative.
Here's my attempt:
We should be able to have a suitably chosen function $f$ on $\mathbf X$ so that $\frac{d}{dt}(S^{(i)}f)(t)=G_if,$ and thus conclude that $\frac{d}{dt}(Sf)(t)=Gf.$ If the generator and semigroup can be defined, then the process exists, yes?
Take two simple processes with gnenerators $G_i$ which are initially in state $0$ and flip to state $i$ at rate $1$. Take these processes, and scale their temporal dynamics by $a$ and $b$ respectively. Then we have a Markov process with generator $G=aG_1+bG_2$, at least for $a,b>0$.
If $X$ has generator $G$ and $b>0$ is a scalar, then $X(bt)$ is a Markov process with generator $bG$.
If $G_1$ and $G_2$ are generators on the same state space, and (writing $D(G_i)$ for the domain of $G_i$) $D(G_1)\cap D(G_2)$ is large enough, then $G_1+G_2$ is the generator of another Markov process. Under the appropriate conditions, the semigroup $S$ associated with $G_1+G_2$ is given by $S(t) =\lim_{n\to\infty}[S^{(1)}(t/n)S^{(2)}(t/n)]^n$. Google Trotter product formula. A good place to learn about such things is the first chapter of Markov Processes:Characterization and Convergence by Ethier and Kurtz.