Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|\_{\infty}$. Let $\Omega\_1$ and $\Omega\_2$ be two Markov generators on $\mathcal{C}(X)$ (using the definition of Liggett) with domains $\mathcal{D}\_1$ and $\mathcal{D}\_2$ such that $\mathcal{D}\_1\cap\mathcal{D}\_2$ is dense in $\mathcal{C}(X)$.


**Question:** Is any positive linear combination of $\Omega\_1$ and $\Omega\_2$ ($\lambda\Omega\_1+\mu\Omega\_2$ where $\lambda,\mu>0$) a Markov generator?


If this is not true in general, is any of these two hypotheses required:


* $(\Omega\_1+\Omega\_2)$ is also a Markov generator
* $\Omega\_1$ and $\Omega\_2$ satisfy both the maximum principle (Proposition 2.2 of Liggett)



Is a linear combination of Markov generator a Markov generator?
The answer to your question is positive under certain conditions. For example, by the "product formula" of H. Trotter, (<https://www.ams.org/journals/proc/1959-010-04/S0002-9939-1959-0108732-6/S0002-9939-1959-0108732-6.pdf>), if $\mathcal D\_1\subset\mathcal D\_2$, then the set of $\mu\ge 0$ such that $\Omega\_1+\mu\Omega\_2$ is a Markov generator is open in $[0,\infty)$ and contains $0$. Trotter provides other sufficient conditions, and several enlightening examples. [As I recall such things are discussed on the book of Ethier & Kurtz, but I do not have my copy at hand.]


