Feynman-Kac Formulae

by Del Moral, Pierre

• ISBN13: 9780387202686
• ISBN10: 0387202684
• Format: Hardcover
• Copyright: 2004-03-30
• Publisher: Springer Nature

Summary

This book contains a systematic and self-contained treatment of Feynman-Kac path measures, their genealogical and interacting particle interpretations, and their applications to a variety of problems arising in statistical physics, biology, and advanced engineering sciences. Topics include spectral analysis of Feynman-Kac-Schr??dinger operators, Dirichlet problems with boundary conditions, finance, molecular analysis, rare events and directed polymers simulation, genetic algorithms, Metropolis-Hastings type models, as well as filtering problems and hidden Markov chains. This text takes readers in a clear and progressive format from simple to recent and advanced topics in pure and applied probability such as contraction and annealed properties of non-linear semi-groups, functional entropy inequalities, empirical process convergence, increasing propagations of chaos, central limit, and Berry Esseen type theorems as well as large deviations principles for strong topologies on path-distribution spaces. Topics also include a body of powerful branching and interacting particle methods and worked out illustrations of the key aspect of the theory. With practical and easy to use references as well as deeper and modern mathematics studies, the book will be of use to engineers and researchers in pure and applied mathematics, statistics, physics, biology, and operation research who have a background in probability and Markov chain theory. From the reviews: "I also recommend this book as informal reading for anyone intersted in the subject, preferably with a strong background in Markov processes; in particular, for someone also familiar with one of the many fields to which the book applies Feynman-Kac models. The book is entertaining and informative." Journal of the American Statistical Association, December 2005

Author Biography

Pierre Del Moral is a research fellow in mathematics at the C.N.R.S. (Centre National de la Recherche Scientifique) at the Laboratoire de Statistique et Probabilites of Paul Sabatier University in Toulouse.

Table of Contents

1 Introduction	1	(46)
1.1 On the Origins of Feynman-Kac and Particle Models	1	(6)
1.2 Notation and Conventions	7	(4)
1.3 Feynman-Kac Path Models	11	(3)
1.3.1 Path-Space and Marginal Models	11	(2)
1.3.2 Nonlinear Equations	13	(1)
1.4 Motivating Examples	14	(15)
1.4.1 Engineering Science	14	(7)
1.4.2 Bayesian Methodology	21	(1)
1.4.3 Particle and Statistical Physics	22	(3)
1.4.4 Biology	25	(3)
1.4.5 Applied Probability and Statistics	28	(1)
1.5 Interacting Particle Systems	29	(8)
1.5.1 Discrete Time Models	30	(4)
1.5.2 Continuous Time Models	34	(3)
1.6 Sequential Monte Carlo Methodology	37	(2)
1.7 Particle Interpretations	39	(2)
1.8 A Contents Guide for the Reader	41	(6)
2 Feynman-Kac Formulae	47	(48)
2.1 Introduction	47	(1)
2.2 An Introduction to Markov Chains	48	(10)
2.2.1 Canonical Probability Spaces	49	(2)
2.2.2 Path-Space Markov Models	51	(1)
2.2.3 Stopped Markov chains	52	(3)
2.2.4 Examples	55	(3)
2.3 Description of the Models	58	(3)
2.4 Structural Stability Properties	61	(7)
2.4.1 Path Space and Marginal Models	62	(1)
2.4.2 Change of Reference Probability Measures	63	(2)
2.4.3 Updated and Prediction Flow Models	65	(3)
2.5 Distribution Flows Models	68	(13)
2.5.1 Killing Interpretation	71	(2)
2.5.2 Interacting Process Interpretation	73	(3)
2.5.3 McKean Models	76	(3)
2.5.4 Kalman-Bucy filters	79	(2)
2.6 Feynman-Kac Models in Random Media	81	(6)
2.6.1 Quenched and Annealed Feynman-Kac Flows	83	(2)
2.6.2 Feynman-Kac Models in Distribution Space	85	(2)
2.7 Feynman-Kac Semigroups	87	(8)
2.7.1 Prediction Semigroups	88	(3)
2.7.2 Updated Semigroups	91	(4)
3 Genealogical and Interacting Particle Models	95	(26)
3.1 Introduction	95	(1)
3.2 Interacting Particle Interpretations	96	(3)
3.3 Particle models with Degenerate Potential	99	(4)
3.4 Historical and Genealogical Tree Models	103	(6)
3.4.1 Introduction	103	(2)
3.4.2 A Rigorous Approach and Related Transport Problems	105	(3)
3.4.3 Complete Genealogical Tree Models	108	(1)
3.5 Particle Approximation Measures	109	(12)
3.5.1 Some Convergence Results	112	(3)
3.5.2 Regularity Conditions	115	(6)
4 Stability of Feynman-Kac Semigroups	121	(36)
4.1 Introduction	121	(1)
4.2 Contraction Properties of Markov Kernels	122	(10)
4.2.1 h-relative Entropy	122	(5)
4.2.2 Lipschitz Contractions	127	(5)
4.3 Contraction Properties of Feynman-Kac Semigroups	132	(14)
4.3.1 Functional Entropy Inequalities	134	(4)
4.3.2 Contraction Coefficients	138	(4)
4.3.3 Strong Contraction Estimates	142	(2)
4.3.4 Weak Regularity Properties	144	(2)
4.4 Updated Feynman-Kac Models	146	(6)
4.5 A Class of Stochastic Semigroups	152	(5)
5 Invariant Measures and Related Topics	157	(30)
5.1 Introduction	157	(3)
5.2 Existence and Uniqueness	160	(1)
5.3 Invariant Measures and Feynman-Kac Modeling	161	(3)
5.4 Feynman-Kac and Metropolis-Hastings Models	164	(2)
5.5 Feynman-Kac-Metropolis Models	166	(21)
5.5.1 Introduction	166	(4)
5.5.2 The Genealogical Metropolis Particle Model	170	(2)
5.5.3 Path Space Models and Restricted Markov Chains	172	(7)
5.5.4 Stability Properties	179	(8)
6 Annealing Properties	187	(28)
6.1 Introduction	187	(2)
6.2 Feynman-Kac-Metropolis Models	189	(8)
6.2.1 Description of the Model	189	(2)
6.2.2 Regularity Properties	191	(2)
6.2.3 Asymptotic Behavior	193	(4)
6.3 Feynman-Kac Trapping Models	197	(18)
6.3.1 Description of the Model	197	(1)
6.3.2 Regularity Properties	198	(3)
6.3.3 Asymptotic Behavior	201	(3)
6.3.4 Large-Deviation Analysis	204	(4)
6.3.5 Concentration Levels	208	(7)
7 Asymptotic Behavior	215	(38)
7.1 Introduction	215	(2)
7.2 Some Preliminaries	217	(4)
7.2.1 McKean Interpretations	218	(1)
7.2.2 Vanishing Potentials	219	(2)
7.3 Inequalities for Independent Random Variables	221	(10)
7.3.1 Lp and Exponential Inequalities	222	(5)
7.3.2 Empirical Processes	227	(4)
7.4 Strong Law of Large Numbers	231	(22)
7.4.1 Extinction Probabilities	231	(5)
7.4.2 Convergence of Empirical Processes	236	(8)
7.4.3 Time-Uniform Estimates	244	(9)
8 Propagation of Chaos	253	(38)
8.1 Introduction	253	(2)
8.2 Some Preliminaries	255	(3)
8.3 Outline of Results	258	(3)
8.4 Weak Propagation of Chaos	261	(1)
8.5 Relative Entropy Estimates	262	(5)
8.6 A Combinatorial Transport Equation	267	(4)
8.7 Asymptotic Properties of Boltzmann-Gibbs Distributions	271	(6)
8.8 Feynman-Kac Semigroups	277	(5)
8.8.1 Marginal Models	278	(2)
8.8.2 Path-Space Models	280	(2)
8.9 Total Variation Estimates	282	(9)
9 Central Limit Theorems	291	(40)
9.1 Introduction	291	(2)
9.2 Some Preliminaries	293	(2)
9.3 Some Local Fluctuation Results	295	(5)
9.4 Particle Density Profiles	300	(6)
9.4.1 Unnormalized Measures	300	(1)
9.4.2 Normalized Measures	301	(2)
9.4.3 Killing Interpretations and Related Comparisons	303	(3)
9.5 A Berry-Esseen Type Theorem	306	(12)
9.6 A Donsker Type Theorem	318	(4)
9.7 Path-Space Models	322	(5)
9.8 Covariance Functions	327	(4)
10 Large-Deviation Principles	331	(56)
10.1 Introduction	333	(6)
10.2 Some Preliminary Results	339	(1)
10.2.1 Topological Properties	339	(1)
10.2.2 Idempotent Analysis	340	(1)
10.2.3 Some Regularity Properties	344	(3)
10.3 Cramer's Method	347	(4)
10.4 Laplace-Varadhan's Integral Techniques	351	(8)
10.5 Dawson-Gartner Projective Limits Techniques	359	(4)
10.6 Sanov's Theorem	363	(1)
10.6.1 Introduction	363	(1)
10.6.2 Topological Preliminaries	364	(1)
10.6.3 Sanov's Theorem in the τ-Topology	370	(4)
10.7 Path-Space and Interacting Particle Models	374	(1)
10.7.1 Proof of Theorem 10.1.1	374	(1)
10.7.2 Sufficient Conditions	376	(1)
10.8 Particle Density Profile Models	377	(1)
10.8.1 Introduction	377	(1)
10.8.2 Strong Large-Deviation Principles	379	(8)
11 Feynman-Kac and Interacting Particle Recipes	387	(40)
11.1 Introduction	387	(2)
11.2 Interacting Metropolis Models	389	(1)
11.2.1 Introduction	389	(1)
11.2.2 Feynman-Kac-Metropolis and Particle Models	390	(1)
11.2.3 Interacting Metropolis and Gibbs Samplers	393	(1)
11.3 An Overview of some General Principles	394	(2)
11.4 Descendant and Ancestral Genealogies	396	(4)
11.5 Conditional Explorations	400	(2)
11.6 State-Space Enlargements and Path-Particle Models	402	(2)
11.7 Conditional Excursion Particle Models	404	(1)
11.8 Branching Selection Variants	405	(1)
11.8.1 Introduction	405	(1)
11.8.2 Description of the Models	408	(1)
11.8.3 Some Branching Selection Rules	409	(1)
11.8.4 Some L2-mean Error Estimates	411	(1)
11.8.5 Long Time Behavior	417	(1)
11.8.6 Conditional Branching Models	419	(1)
11.9 Exercises	420	(7)
12 Applications	427	(96)
12.1 Introduction	427	(2)
12.2 Random Excursion Models	429	(1)
12.2.1 Introduction	429	(1)
12.2.2 Dirichlet Problems with Boundary Conditions	431	(1)
12.2.3 Multilevel Feynman-Kac Formulae	436	(1)
12.2.4 Dirichlet Problems with Hard Boundary Conditions	440	(1)
12.2.5 Rare Event Analysis	444	(1)
12.2.6 Asymptotic Particle Analysis of Rare Events	447	(1)
12.2.7 Fluctuation Results and Some Comparisons	450	(1)
12.2.8 Exercises	453	(6)
12.3 Change of Reference Measures	459	(1)
12.3.1 Introduction	459	(1)
12.3.2 Importance Sampling	460	(1)
12.3.3 Sequential Analysis of Probability Ratio Tests	462	(1)
12.3.4 A Multisplitting Particle Approach	463	(1)
12.3.5 Exercises	465	(4)
12.4 Spectral Analysis of Feynman-Kac-Schrödinger Semigroups	469	(1)
12.4.1 Lyapunov Exponents and Spectral Radii	470	(1)
12.4.2 Feynman-Kac Asymptotic Models	471	(1)
12.4.3 Particle Lyapunov Exponents	473	(1)
12.4.4 Hard, Soft and Repulsive Obstacles	475	(1)
12.4.5 Related Spectral Quantities	477	(1)
12.4.6 Exercises	479	(5)
12.5 Directed Polymers Simulation	484	(1)
12.5.1 Feynman-Kac and Boltzmann-Gibbs Models	484	(1)
12.5.2 Evolutionary Particle Simulation Methods	487	(1)
12.5.3 Repulsive Interaction and Self-Avoiding Markov Chains	488	(1)
12.5.4 Attractive Interaction and Reinforced Markov Chains	490	(1)
12.5.5 Particle Polymerization Techniques	490	(1)
12.5.6 Exercises	495	(2)
12.6 Filtering/Smoothing and Path estimation	497	(1)
12.6.1 Introduction	497	(1)
12.6.2 Motivating Examples	500	(1)
12.6.3 Feynman-Kac Representations	505	(1)
12.6.4 Stability Properties of the Filtering Equations	508	(1)
12.6.5 Asymptotic Properties of Log-likelihood Functions	510	(1)
12.6.6 Particle Approximation Measures	512	(1)
12.6.7 A Partially Linear/Gaussian Filtering Model	513	(1)
12.6.8 Exercises	520	(3)
References	523	(26)
Index	549