Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications

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.

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-Schrödinger 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.

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 Probabilités of Paul Sabatier University in Toulouse. He received his Ph.D. in signal processing at the LAAS-CNRS (Laboratoire d'Analyse et Architecture des Systèmes) of Toulouse. He is one of the principal designers of the modern and recently developing theory on particle methods in filtering theory. He served as a research engineer in thecompany Steria-Digilog from 1992 to 1995 and he has been a visiting professor at Purdue University and Princeton University. He is a former associate editor of the journal Stochastic Analysis and Applications.