Superprocess

A superprocess is a measure-valued Markov process that describes the evolution of a spatially distributed population undergoing random branching. The state at time t is a finite measure X_t on a habitat E (often R^d). Superprocesses arise as scaling limits of branching particle systems when particle mass tends to zero and the population size grows to infinity, yielding a continuous description of mass in space.

A superprocess is specified by two ingredients: the underlying spatial motion and the branching mechanism. The spatial motion is a Markov process on E with generator L, governing how mass moves in space. The branching mechanism ψ is a Laplace exponent of a spectrally positive Lévy process and encodes birth, death, and higher-order splitting rates. The log-Laplace functional of X_t provides a convenient characterization: for a nonnegative test function f, E[exp(-⟨X_t, f⟩) | X_0 = μ] = exp(-⟨μ, u(t, ·)⟩), where u solves the nonlinear partial differential equation ∂_t u = Lu - ψ(u), with initial condition u(0, x) = f(x).

A canonical example is the super-Brownian motion, where L is (1/2)Δ in R^d. In this case, ψ(u) often takes the form αu + βu^2 + ∫(e^{-ur} - 1 + ur 1_{r<1}) ν(dr), with α, β ≥ 0 and ν a Lévy measure; ν = 0 and β > 0 yield quadratic branching. The total mass process is a continuous-state branching process.

Connections to partial differential equations are central: the nonlinear equation ∂_t u = Lu - ψ(u) relates to reaction-diffusion phenomena such as the Fisher–Kolmogorov–Petrovsky–Piskunov equation. Variants include historic and immigration-augmented superprocesses, which extend the basic framework for broader applications.