Continuous time, discrete space

We now move on to general continuous-time Markov processes on countable spaces, i.e., we consider processes \((X_t)_{t>0}\) for which the time variable may assume any value in \([0,\infty )\) (or some bounded interval), and each X(t) takes its values in some countable space \(\mathcal{S}\). While the concept of a Markov transition semigroup introduces itself as a generalization of the one-step transition matrix of a discrete-time process, the task of extracting information on the dynamics of the process now becomes significantly harder. The Kolmogorov equations will be derived by essentially analytical methods, which (as before in the case of the Poisson process) rely on dicretizing the process, which means using skeletons. In order to keep the presentation elementary, some of the results we give are not optimal, and whenever necessary, references are given for sharper results.