Superprocess

Concept in probability theory In probability theory, a superprocess is a measure-valued stochastic process that is usually constructed as a special limit of near-critical branching diffusions. Informally, a superprocess can be seen as a branching process where each particle splits and dies at infinite rates, and evolves in a state space E according to a diffusion equation. We follow the rescaled population of particles, seen as a measure on E. Scaling limit of a discrete branching process Simplest setting Branching Brownian process for N=30 For any integer N ≥ 1 {\displaystyle N\geq 1} , consider a branching Brownian process Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} defined as follows: Start at t = 0 {\displaystyle t=0} with N {\displaystyle N} independent particles distributed according to a probability distribution μ {\displaystyle \mu } . Each particle independently move according to a Brownian motion. Each particle independently dies with rate N {\displaystyle N} . When a particle dies, with probability 1 / 2 {\displaystyle 1/2} it gives birth to two offspring in the same location. The notation Y N ( t , d x ) {\displaystyle Y^{N}(t,dx)} means should be interpreted as: at