continuous-state branching and the Lamperti transform

What is a branching population in the limit of huge size, where the integer count becomes a continuum? The continuous-state branching process (CSBP) is the answer: a nonnegative real-valued Markov process X_t that is the scaling limit of Galton-Watson processes, modelling a 'mass' that branches. Its defining property is the branching property — start with mass x + y and the process is the independent sum of the processes started from x and from y — and it carries the entire branching trichotomy into the continuum. The Lamperti transform is the magical change of time that reveals a CSBP to be, in disguise, a Levy process.

A CSBP is characterized by its branching mechanism psi, a function of Levy-Khintchine type: psi(u) = b u + (1/2) c u^2 + integral_(0,infinity) (e^(-u r) - 1 + u r 1_{r<1}) Pi(dr), where c >= 0 is a Gaussian (continuous branching) coefficient and Pi is a measure on (0, infinity) controlling large reproduction jumps. The Laplace transform of X_t satisfies E[exp(-u X_t) given X_0 = x] = exp(-x v_t(u)), where v_t solves the ODE dv_t/dt = -psi(v_t) — the continuous analogue of iterating the generating function. Lamperti's theorem is the structural gem: every CSBP with mechanism psi can be obtained from a spectrally positive Levy process Y (a Levy process with no negative jumps, whose Laplace exponent is psi) by a random time change. Concretely, if Y is run and time is reparametrized by the integral of the elapsed population, X_t = Y_(theta_t) where theta_t = integral_0^t X_s ds is the cumulative-population clock; equivalently X is the unique solution of a time-changed equation driven by Y, stopped at 0. So branching mass and spectrally-positive Levy processes are the same objects in two clocks.

Why it matters: the Lamperti correspondence turns hard branching questions into Levy-process questions, for which the full machinery of fluctuation theory applies — extinction probabilities become first-passage-to-zero probabilities of Y, and the survival/extinction dichotomy is read off from the sign of psi'(0+) (the Malthusian rate). CSBPs are the population-side counterpart of the genealogical coalescents (the Feller diffusion, mechanism psi(u) = c u^2 /2, is the scaling limit of critical Galton-Watson and the mass process dual to Kingman's coalescent). An honest caveat: the correspondence requires the Levy process to be spectrally positive (no negative jumps) — populations can leap up by a large birth event but mass decreases only continuously, never by a downward jump, which is precisely why the dual Levy process has one-sided jumps. CSBPs with the alpha-stable mechanism psi(u) = c u^alpha (1 < alpha <= 2) are the canonical heavy-reproduction examples and link to the Lambda-coalescents on the genealogy side.