the infinitesimal generator

The infinitesimal generator A of a Markov semigroup is the 'instantaneous rate of change' operator: it captures, in a single object, the entire local dynamics of the process. If the semigroup P_t describes evolution over a finite time, A describes what happens in the first infinitesimal instant; integrating that local rule recovers the whole flow, formally P_t = e^(tA). It answers the question: given the process is at x right now, how fast and in what way does the average of an observable f start to change?

Precisely, A f = lim as t -> 0+ of (P_t f - f) / t, where the limit is taken in the Banach-space norm. The set of f for which this limit exists is the domain D(A), and A is generally an UNBOUNDED operator defined only on that dense domain. Two equivalent readings of the same fact are central: differentiating the semigroup law gives the backward Kolmogorov equation d/dt P_t f = A P_t f = P_t A f (for f in D(A)), so A is literally the time-generator of the flow; and probabilistically, A f(x) = lim (E_x[ f(X_t) ] - f(x)) / t is the expected instantaneous rate of change of f along the process. For a diffusion the generator is a second-order elliptic differential operator, A f = (1/2) sum a_(ij) d^2 f / dx_i dx_j + sum b_i df/dx_i, with b the drift and a = sigma sigma^T the diffusion matrix; for a jump process it is an integral operator A f(x) = integral (f(y) - f(x)) Q(x, dy) built from the jump rates; in general it is an integro-differential operator combining drift, diffusion and jumps (the Lévy-type form).

Why it matters: the generator is the most economical complete description of a Markov process — far more compact than the full transition function — and it is the bridge to PDEs (Kolmogorov/Fokker-Planck), to the martingale problem (f(X_t) - f(X_0) - integral_0^t A f(X_s) ds is a martingale for f in D(A)), and to Dynkin's formula. A persistent caveat: A is only defined on its domain, and you cannot freely apply it to arbitrary functions; many subtle phenomena (boundary behaviour, conservativeness, uniqueness of the process) are encoded precisely in WHICH functions belong to D(A), so a generator is incompletely specified until its domain is given.