1 Introduction.- 1.1 On the Origins of Feynman-Kac and Particle Models.- 1.2 Notation and Conventions.- 1.3 Feynman-Kac Path Models.- 1.3.1 Path-Space and Marginal Models.- 1.3.2 Nonlinear Equations.- 1.4 Motivating Examples.- 1.4.1 Engineering Science.- 1.4.2 Bayesian Methodology.- 1.4.3 Particle and Statistical Physics.- 1.4.4 Biology.- 1.4.5 Applied Probability and Statistics.- 1.5 Interacting Particle Systems.- 1.5.1 Discrete Time Models.- 1.5.2 Continuous Time Models.- 1.6 Sequential Monte Carlo Methodology.- 1.7 Particle Interpretations.- 1.8 A Contents Guide for the Reader.- 2 Feynman-Kac Formulae.- 2.1 Introduction.- 2.2 An Introduction to Markov Chains.- 2.2.1 Canonical Probability Spaces.- 2.2.2 Path-Space Markov Models.- 2.2.3 Stopped Markov chains.- 2.2.4 Examples.- 2.3 Description of the Models.- 2.4 Structural Stability Properties.- 2.4.1 Path Space and Marginal Models.- 2.4.2 Change of Reference Probability Measures.- 2.4.3 Updated and Prediction Flow Models.- 2.5 Distribution Flows Models.- 2.5.1 Killing Interpretation.- 2.5.2 Interacting Process Interpretation.- 2.5.3 McKean Models.- 2.5.4 Kalman-Bucy filters.- 2.6 Feynman-Kac Models in Random Media.- 2.6.1 Quenched and Annealed Feynman-Kac Flows.- 2.6.2 Feynman-Kac Models in Distribution Space.- 2.7 Feynman-Kac Semigroups.- 2.7.1 Prediction Semigroups.- 2.7.2 Updated Semigroups.- 3 Genealogical and Interacting Particle Models.- 3.1 Introduction.- 3.2 Interacting Particle Interpretations.- 3.3 Particle models with Degenerate Potential.- 3.4 Historical and Genealogical Tree Models.- 3.4.1 Introduction.- 3.4.2 A Rigorous Approach and Related Transport Problems.- 3.4.3 Complete Genealogical Tree Models.- 3.5 Particle Approximation Measures.- 3.5.1 Some Convergence Results.- 3.5.2 Regularity Conditions.- 4 Stability of Feynman-Kac Semigroups.- 4.1 Introduction.- 4.2 Contraction Properties of Markov Kernels.- 4.2.1 h-relative Entropy.- 4.2.2 Lipschitz Contractions.- 4.3 Contraction Properties of Feynman-Kac Semigroups.- 4.3.1 Functional Entropy Inequalities.- 4.3.2 Contraction Coefficients.- 4.3.3 Strong Contraction Estimates.- 4.3.4 Weak Regularity Properties.- 4.4 Updated Feynman-Kac Models.- 4.5 A Class of Stochastic Semigroups.- 5 Invariant Measures and Related Topics.- 5.1 Introduction.- 5.2 Existence and Uniqueness.- 5.3 Invariant Measures and Feynman-Kac Modeling.- 5.4 Feynman-Kac and Metropolis-Hastings Models.- 5.5 Feynman-Kac-Metropolis Models.- 5.5.1 Introduction.- 5.5.2 The Genealogical Metropolis Particle Model.- 5.5.3 Path Space Models and Restricted Markov Chains.- 5.5.4 Stability Properties.- 6 Annealing Properties.- 6.1 Introduction.- 6.2 Feynman-Kac-Metropolis Models.- 6.2.1 Description of the Model.- 6.2.2 Regularity Properties.- 6.2.3 Asymptotic Behavior.- 6.3 Feynman-Kac Trapping Models.- 6.3.1 Description of the Model.- 6.3.2 Regularity Properties.- 6.3.3 Asymptotic Behavior.- 6.3.4 Large-Deviation Analysis.- 6.3.5 Concentration Levels.- 7 Asymptotic Behavior.- 7.1 Introduction.- 7.2 Some Preliminaries.- 7.2.1 McKean Interpretations.- 7.2.2 Vanishing Potentials.- 7.3 Inequalities for Independent Random Variables.- 7.3.1 Lp and Exponential Inequalities.- 7.3.2 Empirical Processes.- 7.4 Strong Law of Large Numbers.- 7.4.1 Extinction Probabilities.- 7.4.2 Convergence of Empirical Processes.- 7.4.3 Time-Uniform Estimates.- 8 Propagation of Chaos.- 8.1 Introduction.- 8.2 Some Preliminaries.- 8.3 Outline of Results.- 8.4 Weak Propagation of Chaos.- 8.5 Relative Entropy Estimates.- 8.6 A Combinatorial Transport Equation.- 8.7 Asymptotic Properties of Boltzmann-Gibbs Distributions.- 8.8 Feynman-Kac Semigroups.- 8.8.1 Marginal Models.- 8.8.2 Path-Space Models.- 8.9 Total Variation Estimates.- 9 Central Limit Theorems.- 9.1 Introduction.- 9.2 Some Preliminaries.- 9.3 Some Local Fluctuation Results.- 9.4 Particle Density Profiles.- 9.4.1 Unnormalized Measures.- 9.4.2 Normalized Measures.- 9.4.3 Killing Interpretations and Related Comparisons.- 9.5 A Berry-Esseen Type Theorem.- 9.6 A Donsker Type Theorem.- 9.7 Path-Space Models.- 9.8 Covariance Functions.- 10 Large-Deviation Principles.- 10.1 Introduction.- 10.2 Some Preliminary Results.- 10.2.1 Topological Properties.- 10.2.2 Idempotent Analysis.- 10.2.3 Some Regularity Properties.- 10.3 Crámer¿s Method.- 10.4 Laplace-Varadhan¿s Integral Techniques.- 10.5 Dawson-Gärtner Projective Limits Techniques.- 10.6 Sanov¿s Theorem.- 10.6.1 Introduction.- 10.6.2 Topological Preliminaries.- 10.6.3 Sanov¿s Theorem in the r-Topology.- 10.7 Path-Space and Interacting Particle Models.- 10.7.1 Proof of Theorem 10.1.1.- 10.7.2 Sufficient Conditions.- 10.8 Particle Density Profile Models.- 10.8.1 Introduction.- 10.8.2 Strong Large-Deviation Principles.- 11 Feynman-Kac and Interacting Particle Recipes.- 11.1 Introduction.- 11.2 Interacting Metropolis Models.- 11.2.1 Introduction.- 11.2.2 Feynman-Kac-Metropolis and Particle Models.- 11.2.3 Interacting Metropolis and Gibbs Samplers.- 11.3 An Overview of some General Principles.- 11.4 Descendant and Ancestral Genealogies.- 11.5 Conditional Explorations.- 11.6 State-Space Enlargements and Path-Particle Models.- 11.7 Conditional Excursion Particle Models.- 11.8 Branching Selection Variants.- 11.8.1 Introduction.- 11.8.2 Description of the Models.- 11.8.3 Some Branching Selection Rules.- 11.8.4 Some L2-mean Error Estimates.- 11.8.5 Long Time Behavior.- 11.8.6 Conditional Branching Models.- 11.9 Exercises.- 12 Applications.- 12.1 Introduction.- 12.2 Random Excursion Models.- 12.2.1 Introduction.- 12.2.2 Dirichlet Problems with Boundary Conditions.- 12.2.3 Multilevel Feynman-Kac Formulae.- 12.2.4 Dirichlet Problems with Hard Boundary Conditions.- 12.2.5 Rare Event Analysis.- 12.2.6 Asymptotic Particle Analysis of Rare Events.- 12.2.7 Fluctuation Results and Some Comparisons.- 12.2.8 Exercises.- 12.3 Change of Reference Measures.- 12.3.1 Introduction.- 12.3.2 Importance Sampling.- 12.3.3 Sequential Analysis of Probability Ratio Tests.- 12.3.4 A Multisplitting Particle Approach.- 12.3.5 Exercises.- 12.4 Spectral Analysis of Feynman-Kac-Schrödinger Semigroups.- 12.4.1 Lyapunov Exponents and Spectral Radii.- 12.4.2 Feynman-Kac Asymptotic Models.- 12.4.3 Particle Lyapunov Exponents.- 12.4.4 Hard, Soft and Repulsive Obstacles.- 12.4.5 Related Spectral Quantities.- 12.4.6 Exercises.- 12.5 Directed Polymers Simulation.- 12.5.1 Feynman-Kac and Boltzmann-Gibbs Models.- 12.5.2 Evolutionary Particle Simulation Methods.- 12.5.3 Repulsive Interaction and Self-Avoiding Markov Chains.- 12.5.4 Attractive Interaction and Reinforced Markov Chains.- 12.5.5 Particle Polymerization Techniques.- 12.5.6 Exercises.- 12.6 Filtering/Smoothing and Path estimation.- 12.6.1 Introduction.- 12.6.2 Motivating Examples.- 12.6.3 Feynman-Kac Representations.- 12.6.4 Stability Properties of the Filtering Equations.- 12.6.5 Asymptotic Properties of Log-likelihood Functions.- 12.6.6 Particle Approximation Measures.- 12.6.7 A Partially Linear/Gaussian Filtering Model.- 12.6.8 Exercises.- References.

The central theme of this book concerns Feynman-Kac path distributions, interacting particle systems, and genealogical tree based models. This re­ cent theory has been stimulated from different directions including biology, physics, probability, and statistics, as well as from many branches in engi­ neering science, such as signal processing, telecommunications, and network analysis. Over the last decade, this subject has matured in ways that make it more complete and beautiful to learn and to use. The objective of this book is to provide a detailed and self-contained discussion on these connec­ tions and the different aspects of this subject. Although particle methods and Feynman-Kac models owe their origins to physics and statistical me­ chanics, particularly to the kinetic theory of fluid and gases, this book can be read without any specific knowledge in these fields. I have tried to make this book accessible for senior undergraduate students having some familiarity with the theory of stochastic processes to advanced postgradu­ ate students as well as researchers and engineers in mathematics, statistics, physics, biology and engineering. I have also tried to give an "expose" of the modem mathematical theory that is useful for the analysis of the asymptotic behavior of Feynman-Kac and particle models.