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Title

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Contents

Preface

Chapter 1. Superprocesses as Diffusion Approximations

Summary

The Dawson-Wat anabe superprocess

1.1. Branching Brownian motion

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

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

Chapter 2. Qualitative Behaviour I

Summary

The Dawson-Watanabe superprocess via its dual

2.1. A series solution to the evolution equation

2.2. Moments of the Dawson-Wat anabe superprocess

2.3. The density in one dimension

2.4. The spde viewpoint

2.5. Occupation times

2.6. Continuity and discontinuity

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)

3.3. The Evans Immortal Particle

3.4. Other skeletons

3.5. Le Jan's construction

3.6. The Brownian Snake

3.7. The infinite variance snake

3.8. Superprocesses and subordination

Chapter 4. The Relationship Between Our Two Classes of Superprocess

Summary

4.1. Approximating particle systems revisited

4.2. Generators revisited

4.3. The generator in polar coordinates

4.4. Consequences of the polar form of the generator

Chapter 5. A Countable Representation

Summary

5.1. A second look at the moment equations for Fleming-Viot

5.2. The lookdown process

5.3. The modified Donnelly-Kurtz construction

5.4. Incorporating selection

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-Wat anabe superprocess

6.5. Charging and hitting sets

6.6. Intersection and collision local times

Chapter 7. Introducing Interactions

Summary

7.1. The basic superprocesses as building blocks

Perkins' stochastic calculus

7.2. A discrete approximation

7.3. A countable representation

7.4. Dawson's Girsanov transform

7.5. Nonlinear branching

7.6. The Fleming-Viot process with selection

7.7. A bivariate Girsanov transform

Chapter 8. Superprocesses and Partial Differential Equations

Summary

8.1. Exit measure

8.2. Solving a nonlinear Dirichlet problem

8.3. Polar sets and capacity

8.4. Le Gall's representation of solutions and related results

8.5. Super-regularity of boundary points

Chapter 9. Some More Interacting Models

Summary

9.1. Multi-level superprocesses

9.2. Competing species superprocesses

9.3. Rescaling the contact process

9.4. Catalytic Superprocesses

9.5. Mutually catalytic superprocesses

Appendix

A.1. Kurtz's tightness criterion

A.2. Ordered binary rooted trees

A.3. White noise

A.4. Gronwall's inequality

Bibliography

Index of Notation

Index

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B

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D

E

F

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L

M

N

O

P

Q

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U

V

W

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