Classical and Spatial Stochastic Processes

Abstract

I Discrete Time Markov Chains.- I.1 Three fundamental examples.- I.2 Classification of states.- I.3 Finite Markov chains.- I.4 Birth and death chains.- I.5 An example of coupling.- I.6 Time to get ruined.- I.7 Absorption probabilities for martingales.- I.8 Random walks.- I.9 The Bienayme-Galton-Watson branching process.- I.10 Proof of Theorem I.2.1.- I.11 Proof of Theorem I.2.2.- I.12 Proof of Theorem I.9.1.- II Stationary Distributions of a Markov Chain.- II. 1 Existence of stationary distributions.- II.2 Reversible measures.- II.3 Convergence to a stationary distribution.- II.4 The finite case.- II.5 Proof of Proposition II. 1.2.- II.6 Proof of Proposition II. 1.3.- II.7 Proofs of Theorems II.3.1 and II.4.2.- III Continuous Time Birth and Death Markov Chains.- III.1 The exponential distribution.- III.2 Construction and first properties.- III.3 Limiting probabilities.- III.4 Classification of states.- III.5 The Poisson process.- III.6 Passage times.- III.7 A queue which is not Markovian.- III.8 Proof of Theorem III.3.1.- III.9 Proof of Theorem III.5.1.- III.10 Proof of Theorem III.5.2.- IV Percolation.- IV.1 Percolation on Zd.- IV.2 Further properties of percolation on Zd.- IV.3 Percolation on a tree and two critical exponents.- V A Cellular Automaton.- V.1 The model.- V.2 A renormalization argument.- VI Continuous Time Branching Random Walk.- VI.1 A continuous time Bienayme-Galton-Watson process.- VI.2 A continuous time branching random walk.- VI.3 The first phase transition is continuous.- VI.4 The second phase transition is discontinuous.- VI.5 Proof of Theorem VI.2.1.- VII The Contact Process on a Homogeneous Tree.- VII.1 The two phase transitions.- VII.2 Characterization of the first phase transition.- VII.3 The graphical construction.- VII.4 Proofs.- VII.5 Open problems.- Appendix: Some Facts About Probabilities on Countable Spaces.- 1 Probability space.- 2 Independence.- 3 Discrete random variables.- References.