Introduction

Abstract

Over the years I have heard a number of complaints about the impenetrable literature on measure-valued branching processes or Dawson-Watanabe superprocesses. These concerns have in part been addressed by some recent publications including Don Dawson–s St. Flour notes (Dawson (1993)), Eugene Dynkin–s monograph (Dynkin (1994)) and Jean-Francois Le Gall–s ETH Lecture Notes (Le Gall (1999)). Nonetheless, one still hears that several topics are only accessible to experts. However, each time I asked a colleague what topics they would like to see treated in these notes, I got a different suggestion. Although there are some other less flattering explanations, I would like to thick the lack of a clear concensus is a reflection of the large number of different entry points to the subject. The Fleming-Viot processes, used to model genotype frequencies in population genetics, arise by conditioning the total mass of a superprocess to be one (Etheridge and March (1991)). When densities exist (as for super-Brownian motion in one spatial dimension) they typically are solutions of parabolic stochastic pde–s driven by a white noise and methods developed for their study often have application to large classes of stochastic pde–s (e.g. Mueller and Perkins (1992), Krylov (1997b), Mytnik (1998) and Section III.4). Dawson-Watanabe superprocesses arise as scaling limits of interacting particle systems (Cox, Durrett and Perkins (1999, 2000)) and of oriented percolation at criticality (recent work of van der Hofstad and Slade (2000)). Rescaled lattice trees above eight dimensions converge to the integral of the super-Brownian cluster conditioned to have mass one (Derbez and Slade (1998)). There are close connections with class of nonlinear pde–s and the interaction between these fields had led to results for both (Dynkin and Kuznetsov (1996, 1998), Le Gall (1999) and Section III.5). They provide a rich source of exotic path properties and an interesting collection of random fractals which are amenable to detailed study (Perkins (1988), Perkins and Taylor (1998), and Chapter III).