On the differential equations for the transition probabilities of Markov processes with enumerably many states

Let p ik ( s, t ) ( i, k = 1, 2, …; s ≤ t ) be the transition probabilities of a Markov process in a system with an enumerable set of states. The states are labelled by positive integers, and p ik ( s, t ) is the conditional probability that the system be in state k at time t, given that it was in state i at an earlier time s. If certain regularity conditions are imposed on the p ik , they can be shown to satisfy the well-known Kolmogorov equations §