Superprocess

An $(\alpha,d,\beta)$-superprocess, $X(t,dx)$, is a stochastic process on $\mathbb{R} \times \mathbb{R}^d$ that is usually constructed as a special limit of branching diffusion where the branching mechanism is given by its factorial moment generating function:

$\Phi(s) = \frac{1}{1+\beta}(1-s)^{1+\beta}+s$

and the spatial motion of individual particles is given by the $\alpha$-symmetric stable process with infinitesimal generator $\Delta_{\alpha}$.

The $\alpha = 2$ case corresponds to standard Brownian motion and the $(2,d,1)$-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

$\Delta u-u^2=0\ on\ \mathbb{R}^d.$