In a continuous-time Markov chain with state X(t) at time t, define p_ij(t) as the probability that X(t + s) = j, given that X(s) = i, s, t ≥ 0, and r_ij as the transition rate out of state i to state j. Then Kolmogorov's backward equations say that, for all states i, j and times t ≥ 0, the derivatives dp_ij(t)/dt = Σ_k≠ir_ikp_kj(t) − v_ip_ij(t), where v_i is the transition rate out of state i, v_i = Σ_jr_ij.  Markov chains;  Markov processes.