Superprocess - Wikipedia

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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}.

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.

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