An Introduction to Superprocesses

Abstract

Preface vii Chapter 1. Superprocesses as Diffusion Approximations Summary The Dawson-Watanabe superprocess 1.1. Branching Brownian motion 1 1.2. A martingale characterisation 1.3. The Feller diffusion 1.4. Rescaling and tightness 1.5. The Dawson-Watanabe martingale problem 1.6. The method of duality 1.7. A more general class of superprocesses 1.8. Infinite initial measures 1.9. Historical superprocesses 24 The Fleming-Viot superprocess 1.10. The stepwise mutation model 1.11. The Fleming-Viot martingale problem 1.12. A dual process for Fleming-Viot 29 Chapter 2. Qualitative Behaviour I Summary The Dawson-Watanabe superprocess via its dual 2.1. A series solution to the evolution equation 33 2.2. Moments of the Dawson-Watanabe superprocess 37 2.3. The density in one dimension 2.4. The spde viewpoint 2.5. Occupation times 44 2.6. Continuity and discontinuity 47 2.7. The extinction/persistence dichotomy The Fleming-Viot superprocess: first properties 2.8. Moments 2.9. The density in one dimension Chapter 3. The Le Gall Representation Summary 3.1. The branching process in random walk 3.2. The Feller rescaling (again) 59 3.3. The Evans Immortal Particle 3.4. Other skeletons 3.5. Le Jan's construction iii iv CONTENTS 3.6. The Brownian Snake 67 3.7. The infinite variance snake 3.8. Superprocesses and subordination 72 Chapter 4. The Relationship Between Our Two classes of Superprocess 77 Summary 77 4.1. Approximating particle systems revisited 77 4.2. Generators revisited 78 4.3. The generator in polar coordinates 80 4.4. Consequences of the polar form of the generator Chapter 5. A Countable Representation 85 Summary 85 5.1. A second look at the moment equations for Fleming-Viot 85 5.2. The lookdown process 86 5.3. The modified Donnelly-Kurtz construction 90 5.4. Incorporating selection 95 5.5. Some old results revisited Chapter 6. Qualitative Behaviour II Summary 6.1. Cluster representations 6.2. The historical modulus of continuity 6.3. The Hausdorff measure of the support 6.4. Palm distributions for the Dawson-Watanabe superprocess 6.5. Charging and hitting sets 6.6. Intersection and collision local times Chapter 7.

A master equation for a spatial population model with pair interactions

“…1 grow larger and farther apart with increasing time (see also Young et al, 2001). Mathematicians refer to this special case nðrÞ ¼ 0 of the Brownian bug model as ''super-Brownian motion'' (Adler, 1997;Etheridge, 2000;Slade, 2002). The unrestrained clumping produced with nðrÞ ¼ 0 has been independently studied by statistical physicists (Zhang et al, 1990;Meyer et al, 1996;Kessler et al, 1997).…”

Infinite Canonical Super-Brownian Motion and Scaling Limits

“…The r-point functions appear explicitly in this description, since 21) where (x j , n j ) equals (y i , m i ) precisely a i times. Thus, the joint moments of the measures {µ n } ∞ n=0 are equal to the r-point functions.…”

Measure-valued limits of interacting particle systems with k-nary interactions II. Finite-dimensional limits

“…Ref. [10] for the case of superprocesses. Alternatively, even simpler, one deduces the tightness directly from Remark 5.2 in Chapter 4 of Ref.…”

“…Refs. [9,10]), let us say that the interaction of particles of d types is local, if a group of particles specified by a profile C can produce particles only of type j [ supp C. Processes subject to this restriction include a variety of the so-called birth and death processes from the theory of multidimensional population processes (see, e.g. Ref.…”

“…Refs. [9,10,23] and references therein), one considers points on Z d þ as integer -valued measures on {1, . .…”