内容

目次

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.