**University Lecture Series**

Volume: 20;
2000;
187 pp;
Softcover

MSC: Primary 60;

Print ISBN: 978-0-8218-2706-2

Product Code: ULECT/20

List Price: $41.00

Individual Member Price: $32.80

**Electronic ISBN: 978-1-4704-2167-0
Product Code: ULECT/20.E**

List Price: $41.00

Individual Member Price: $32.80

# An Introduction to Superprocesses

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*Alison M. Etheridge*

Over the past 20 years, the study of superprocesses has expanded into a major
industry and can now be regarded as a central theme in modern probability
theory. This book is intended as a rapid introduction to the subject, geared
toward graduate students and researchers in stochastic analysis.

A variety of different approaches to the superprocesses emerged over the last
ten years. Yet no one approach superseded any others. In this book, readers are
exposed to a number of different ways of thinking about the processes, and each
is used to motivate some key results. The emphasis is on why results are true
rather than on rigorous proof. Specific results are given, including extensive
references to current literature for their general form.

#### Readership

Graduate students and research mathematicians interested in probability and its applications.

#### Reviews & Endorsements

The book is well written … makes for joyful reading. The style is partially informal and aims more at an intuitive understanding than at technical proofs … The book is thus very suitable for those who want rapid and comprehensive information and are ready to go back to the original works for technical details.

-- Translated from Jahresbericht der Deutschen Mathematiker-Vereinigung

#### Table of Contents

# Table of Contents

## An Introduction to Superprocesses

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1. Superprocesses as Diffusion Approximations 114 free
- Summary 114
- The Dawson-Wat anabe superprocess 114
- 1.1. Branching Brownian motion 114
- 1.2. A martingale characterisation 417
- 1.3. The Feller diffusion 720
- 1.4. Rescaling and tightness 821
- 1.5. The Dawson-Watanabe martingale problem 1326
- 1.6. The method of duality 1629
- 1.7. A more general class of superprocesses 1730
- 1.8. Infinite initial measures 2235
- 1.9. Historical superprocesses 2437
- The Fleming-Viot superprocess 2538
- 1.10. The stepwise mutation model 2538
- 1.11. The Fleming-Viot martingale problem 2740
- 1.12. A dual process for Fleming-Viot 2942

- Chapter 2. Qualitative Behaviour I 3346
- Summary 3346
- The Dawson-Watanabe superprocess via its dual 3346
- 2.1. A series solution to the evolution equation 3346
- 2.2. Moments of the Dawson-Wat anabe superprocess 3750
- 2.3. The density in one dimension 4053
- 2.4. The spde viewpoint 4255
- 2.5. Occupation times 4457
- 2.6. Continuity and discontinuity 4760
- 2.7. The extinction/persistence dichotomy 4962
- The Fleming-Viot superprocess: first properties 5568
- 2.8. Moments 5568
- 2.9. The density in one dimension 5669

- Chapter 3. The Le Gall Representation 5770
- Summary 5770
- 3.1. The branching process in random walk 5770
- 3.2. The Feller rescaling (again) 5972
- 3.3. The Evans Immortal Particle 6073
- 3.4. Other skeletons 6376
- 3.5. Le Jan's construction 6477
- 3.6. The Brownian Snake 6780
- 3.7. The infinite variance snake 6982
- 3.8. Superprocesses and subordination 7285

- Chapter 4. The Relationship Between Our Two Classes of Superprocess 7790
- Chapter 5. A Countable Representation 8598
- Chapter 6. Qualitative Behaviour II 105118
- Summary 105118
- 6.1. Cluster representations 106119
- 6.2. The historical modulus of continuity 108121
- 6.3. The Hausdorff measure of the support 111124
- 6.4. Palm distributions for the Dawson-Wat anabe superprocess 116129
- 6.5. Charging and hitting sets 118131
- 6.6. Intersection and collision local times 121134

- Chapter 7. Introducing Interactions 125138
- Summary 125138
- 7.1. The basic superprocesses as building blocks 125138
- Perkins' stochastic calculus 127140
- 7.2. A discrete approximation 127140
- 7.3. A countable representation 129142
- 7.4. Dawson's Girsanov transform 133146
- 7.5. Nonlinear branching 138151
- 7.6. The Fleming-Viot process with selection 139152
- 7.7. A bivariate Girsanov transform 140153

- Chapter 8. Superprocesses and Partial Differential Equations 143156
- Chapter 9. Some More Interacting Models 157170
- Appendix 173186
- Bibliography 177190
- Index of Notation 183196 free
- Index 185198
- Back Cover Back Cover1201