the transition semigroup

The transition semigroup is the family of linear operators (P_t)_(t >= 0) through which a Markov process acts on functions of its state. It is the central object of the operator approach: instead of tracking sample paths, we watch how observables evolve. Define (P_t f)(x) = E_x[ f(X_t) ] = integral f(y) p_t(x, dy) — the expected value of the observable f, t units of time after starting deterministically at x. As t and x vary, P_t f is a new function, and the whole theory studies the dynamics of f -> P_t f.

It is a semigroup because Chapman-Kolmogorov gives P_(t+s) = P_t P_s with P_0 = I. Three structural facts make it analytically tractable. First, P_t is POSITIVITY-PRESERVING and a Markov operator: f >= 0 implies P_t f >= 0, and P_t 1 = 1 (constants are preserved — total probability is conserved). Second, on a suitable Banach space of functions (often the continuous functions vanishing at infinity, C_0(E), or an L^p(mu) space) it is a CONTRACTION: the sup-norm or L^p-norm of P_t f never exceeds that of f, because averaging cannot increase a maximum. Third, with the right space and a continuity hypothesis it is STRONGLY CONTINUOUS, meaning P_t f -> f as t -> 0. These three properties are what let us define the generator and invoke Hille-Yosida.

Why it matters: every probabilistic question becomes an operator question. Stationary measures are the mu with mu P_t = mu (left eigen-measure with eigenvalue 1); the generator A is the t-derivative at 0; the backward Kolmogorov equation is d/dt P_t f = A P_t f; reversibility is self-adjointness of P_t in L^2(mu). The choice of Banach space is not cosmetic: a semigroup may be strongly continuous on C_0(E) (Feller) but only weak-* continuous on the space of bounded measurable functions, and the generator's domain depends on the space, so one must state on which space one is working.