the generator of a Levy process
/ lev-EE /
The generator of a Lévy process is the operator that describes its infinitesimal evolution — the time derivative at zero of the transition semigroup P_t f(x) = E[f(x + X_t)]. For a Lévy process this generator has a striking dual face: in Fourier space it is multiplication by the characteristic exponent psi, making it a pseudo-differential operator; in real space it is an integro-differential operator, a second-order differential part plus an integral over jumps.
Concretely, for smooth compactly supported f the generator is A f(x) = b f'(x) + (1/2) sigma^2 f''(x) + integral over R\{0} of (f(x + y) - f(x) - y f'(x) 1_{|y|<1}) nu(dx). Read it off the Lévy triplet: b f' is the drift transport, (sigma^2/2) f'' is the Brownian diffusion (a second-order elliptic term), and the integral is the jump part — it compares f at x+y to f at x for every possible jump size y, weighted by the Lévy measure, with the small-jump compensation -y f'(x) ensuring convergence near 0 exactly as in Lévy-Khintchine. Taking Fourier transforms, A becomes the multiplier psi: the Fourier transform of A f is psi(theta) times the Fourier transform of f, so A is the pseudo-differential operator with symbol psi. The semigroup is then formally P_t = e^(tA), and the law of X_t evolves by the (generalized, nonlocal) Kolmogorov forward equation generated by A.
The generator is the bridge from Lévy processes to PDE and to general Markov theory: it places Lévy processes inside the Feller class, gives the Dynkin formula and the martingale problem, and turns probabilistic questions into the analysis of a nonlocal operator (the fractional Laplacian -(-Delta)^(alpha/2) is exactly the generator of the symmetric alpha-stable process). The caveat: this operator is NONLOCAL — A f(x) depends on the values of f far from x through the jump integral — so the partial integro-differential equations of jump processes are genuinely different from the local PDEs of diffusions, and intuition from second-order elliptic equations does not transfer wholesale.
For Brownian motion (b=0, sigma=1, nu=0) the generator is A = (1/2) d^2/dx^2, the classical heat operator. For the symmetric alpha-stable process (sigma=0, nu(dx)=c|x|^(-1-alpha)dx) the generator is the fractional Laplacian A = -c(-Delta)^(alpha/2), a nonlocal operator whose symbol is -c|theta|^alpha = psi(theta) — the jump structure becomes a fractional derivative.
Brownian -> Laplacian; alpha-stable -> fractional Laplacian.
The generator is nonlocal whenever there are jumps: A f(x) involves f at points far from x. Its Fourier symbol is exactly the characteristic exponent psi, which is why the same psi controls both the law (via e^(t psi)) and the dynamics (via A).