Superprocess

An -superprocess, $X(t,dx)$, is a stochastic process on $mathbb\{R\}\; imes\; mathbb\{R\}^d$ that is usually constructed as a special limit of branching diffusion where the branching mechanism is given by its factorial moment generating function:: and the spatial motion of individual particles is given by the $alpha$-symmetric stable process with infinitessimal 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:$Delta\; u-u^2=0\; on\; mathbb\{R\}^d.$

References

*cite book | author=Eugene B. Dynkin | year = 2004
title = Superdiffusions and positive solutions of nonlinear partial differential equations. Appendix A by J.-F. Le Gall and Appendix B by I. E. Verbitsky
publisher = University Lecture Series, 34. American Mathematical Society

*cite book | author=Alison Etheridge | year = 2000
title = An Introduction to Superprocesses
publisher = American Mathematical Society