Chapman-kolmogorov equations

In a parameter-homogeneous Markov chain {X(t)} with state space S, define p _ij (t) as the probability that X(t + s) = j, given that X(s) = i for s, t ≥ 0. Then, for all states i, j and index parameters s, t ≥ 0,

are the Chapman-Kolmogorov equations. There is a comparable definition when the state space is instead continuous. See Markov chains; Markov processes.