Summary: Liggett's 1985 and 1999, are the bible (old and new testament), but as Lanchier wrote in his 2024 book there is still room for another treatment. This book is sort of my math autobiogrphy, since it will feature a significant fraction of my work on the subject. As the end of the title indicates, the book will concentrate on the ideas that go into the proofs and highlight the most important techniques: duality, the block construction, and the convergence of rescaled particle systems to ODE, PDE, and IDE. Applications to systems in ecology, genetics, and evolutionary games will illustrate the use of the theory. Applications are mostly at a conceptual level, i.e., identifying the features of a system that are responsible for its observed behavior.
Nov 19, 2025 build - 246 pages
1.1. A few words about Markov processes
1.2. Harris construction for finite range models
1.3. Additive processes, graphical representations
1.4. Attractive processes
1.5. Properties of the voter model and CRW
1.6. Oriented percolation
1.7. Properties of the contact process
2.1. A simple prototype
2.2. Long range contact process
2.3. Models with a dominant species
2.4. Threshold models
2.5. Interlude: Weinberger (1982)
2.6. Quadratic contact process
2.7. Staver-Levin forest model
3.1. Competing contact processes
3.2. Heuristics of Durrett and Levin (1994)
3.3. Mutual invadability implies coexistence
3.4. Lotka-Volterra systems
3.5. Rock-paper-scissors
4.1. Four simple examples
4.2. From heuristics to proofs
4.3. Predator-prey systems
4.4. Catalytic surfaces
4.5. Mutual invadability implies coexistence, II
4.6. Lyapunov functions for Lotka-Volterra systems
4.7. Two prey, one predator
5.1. Basic theory
5.2. Hydrodynamic limit
5.3 General coexistence and extinction results
5.4. Limiting behavior of PDE
5.5. Lotka-Volterra systems
5.6. Evolution of cooperation
5.7. General 2 by 2 games
5.8. Three strategy games
5.9. A public goods game in pancreatic cancer
5.10. Contact process with fast voting
6.1. Voter model perturbations on the torus
6.2. Latent voter model
6.3. q-voter model
6.4. Darling and Norris (2008)
6.5. Molofsky et al (1999)
6.6. Interlude: Ising models
6.7. Motion by mean curvature
6.8. A nonergodic particle system with positive rates?
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