 Solution

A conservative Q-matrix has $q_{ij}\ge 0$ for $i\ne j$, finite exit rates $-q_{ii}={\sum }_{j\ne i}{q}_{ij}$, and hence row sums zero. It specifies the infinitesimal transition probabilities $P_{ij}(h)={\delta }_{ij}+h{q}_{ij}+o(h)$. For a finite state space,The first derivative at zero is the generator. On infinite state spaces one must additionally address nonexplosion and the domains needed for the forward and backward equations.