An -superprocess, , is a stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 on that is usually constructed as a special limit of branching diffusion where the branching mechanism is given by its factorial moment generating function:
and the spatial motion of individual particles is given by the -symmetric stable process

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

 with infinitesimal generator
.

The case corresponds to standard Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 and the -superprocess is called the Dawson-Watanabe superprocess or super-Brownian motion.

One of the most important properties of superprocesses is that they are intimately connected with certain nonlinear partial differential equations.
The simplest such equation is