the second-order elliptic generator of a diffusion

Every diffusion has an infinitesimal generator L, the operator that captures its dynamics in a single differential expression. For the diffusion solving dX = b(X) dt + sigma(X) dB, the generator is the second-order partial differential operator L f(x) = sum_i b_i(x) partial_i f(x) + (1/2) sum_{i,j} a_ij(x) partial_i partial_j f(x), where a = sigma sigma^T. It is the diffusion's calling card: from L you can read off the drift (the first-order part) and the diffusion matrix (the second-order part), and L determines the whole law of the process. The question it answers is: what single operator encodes 'where the process is heading and how it spreads'?

L is the infinitesimal generator of the transition semigroup P_t f(x) = E[f(X_t) given X_0 = x] in the Markov-semigroup sense: L f = lim_{t -> 0} (P_t f - f)/t, on the appropriate domain. By Ito's formula, for smooth f the process f(X_t) - f(X_0) - integral_0^t L f(X_s) ds = integral_0^t (grad f)^T sigma dB is a local martingale — this is Dynkin's formula in differential form and is exactly the martingale-problem characterization. The operator is elliptic because the second-order coefficient matrix a = sigma sigma^T is symmetric and nonnegative (positive semidefinite); it is uniformly/strictly elliptic when a >= delta I for some delta > 0, i.e. the noise is nondegenerate in every direction. The first-order term b is the drift / advection, the second-order term (1/2) a : D^2 is the diffusion / spreading, and there is no zeroth-order term unless the process is killed.

The generator is the hinge connecting probability to analysis. The backward Kolmogorov equation is partial_t u = L u (run forward in t, with L acting on the spatial variable); the Fokker-Planck (forward) equation is partial_t p = L^* p with the formal adjoint L^*; Feynman-Kac adds a potential term to L; the invariant density solves L^* p = 0. Honest cautions: 'the generator' is really an operator with a carefully specified domain (Hille-Yosida theory), and on that domain it is closed and densely defined — you cannot ignore the domain when stating, e.g., the resolvent or self-adjointness. Ellipticity that degenerates (a not strictly positive) means the diffusion lives on a sub-manifold and the density need not be smooth; Hormander's condition on the brackets of the vector fields is the precise repair (hypoellipticity).