Superprocess

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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[edit]

  • 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.