# Superprocess

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

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

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

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

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

## References

• Eugene B. Dynkin (2004). Superdiffusions and positive solutions of nonlinear partial differential equations. Appendix A by J.-F. Le Gall and Appendix B by I. E. Verbitsky. University Lecture Series, 34. American Mathematical Society. ISBN 9780821836828.
• Alison Etheridge (2000). An Introduction to Superprocesses. American Mathematical Society. ISBN 9780821827062.