the Chapman-Kolmogorov equation as the semigroup property
/ CHAP-man kol-mo-GO-rov /
The Chapman-Kolmogorov equation is the single consistency law a Markov transition function must obey, and it is the source of the entire operator-semigroup machinery. In words: to travel from x and land in B after time t+s, you must be SOMEWHERE at the intermediate time t, then move on for the remaining time s. Summing (integrating) over all intermediate positions reconstructs the long transition from two shorter ones. This is not an extra assumption layered on the Markov property; it is the Markov property written for the transition function.
Written with kernels, p_(t+s)(x, B) = integral over E of p_s(y, B) p_t(x, dy) for all t, s >= 0. The crucial reinterpretation is operatorial. Define the transition operators (P_t f)(x) = integral f(y) p_t(x, dy). Then Chapman-Kolmogorov becomes the clean algebraic identity P_(t+s) = P_t P_s, with P_0 = I (the identity). A family of operators satisfying P_(t+s) = P_t P_s and P_0 = I is a one-parameter semigroup — the same structure as the exponential law e^((t+s)a) = e^(ta) e^(sa) for numbers. Thus 'Chapman-Kolmogorov' and 'the (P_t) form a semigroup' are two names for one fact, and they explain why we will eventually be able to write P_t = e^(tA) for a generator A.
Why it matters: the semigroup viewpoint converts probabilistic dynamics into linear algebra/analysis on a function space, where the generator A = d/dt P_t at t = 0 becomes the central object and PDEs (the Kolmogorov equations) drop out by differentiating P_(t+s) = P_t P_s. A subtlety worth stating: composition of operators is generally NOT commutative, yet here P_t P_s = P_s P_t = P_(t+s) because they are powers of the same one-parameter flow; this commutativity in time is exactly what 'time-homogeneous' buys you, and it fails for time-inhomogeneous (explicitly t-dependent) dynamics.
For Brownian motion, plugging the Gaussian kernels into p_(t+s)(x, z) = integral p_t(x, y) p_s(y, z) dy is just the statement that the convolution of an N(0, t) and an N(0, s) density is an N(0, t+s) density — variances add. That is Chapman-Kolmogorov P_(t+s) = P_t P_s made concrete.
Variances adding under convolution is the Chapman-Kolmogorov semigroup law for the Gaussian (heat) semigroup.
Time-homogeneity is essential: the clean law P_(t+s) = P_t P_s holds only when transition probabilities depend on the elapsed time, not the calendar time.