Mean-field particle methods are a broad class of interacting type Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states. A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on sequential interacting samples. The terminology mean-field reflects the fact that each of the samples (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. In other words, starting with a chaotic configuration based on independent copies of initial state of the nonlinear Markov chain model, the chaos propagates at any time horizon as the size the system tends to infinity; that is, finite blocks of particles reduces to independent copies of the nonlinear Markov process. This result is called the propagation of chaos property. The terminology "propagation of chaos" originated with the work of Mark Kac in 1976 on a colliding mean-field kinetic gas model.
History The theory of mean-field interacting particle models had certainly started by the mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. The mathematical foundations of these classes of models were developed from the mid-1980s to the mid-1990s by several mathematicians, including Werner Braun, Klaus Hepp, Karl Oelschläger, Gérard Ben Arous and Marc Brunaud, Donald Dawson, Jean Vaillancourt and Jürgen Gärtner, Christian Léonard, Sylvie Méléard, Sylvie Roelly, Alain-Sol Sznitman and Hiroshi Tanaka for diffusion type models; F. Alberto Grünbaum, Tokuzo Shiga, Hiroshi Tanaka, Sylvie Méléard and Carl Graham for general classes of interacting jump-diffusion processes. We also quote an earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, using mean-field but heuristic-like genetic methods for estimating particle transmission energies. Mean-field genetic type particle methods are also used as heuristic natural search algorithms (a.k.a. metaheuristic) in evolutionary computing. The origins of these mean-field computational techniques can be traced to 1950 and 1954 with the work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey. The Australian geneticist Alex Fraser also published in 1957 a series of papers on the genetic type simulation of artificial selection of organisms. Quantum Monte Carlo, and more specifically Diffusion Monte Carlo methods can also be interpreted as a mean-field particle approximation of Feynman-Kac path integrals. The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948 a mean field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984 In molecular chemistry, the use of genetic heuristic-like particle methods (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with the seminal work of Marshall. N. Rosenbluth and Arianna. W. Rosenbluth. The first pioneering articles on the applications of these heuristic-like particle methods in nonlinear filtering problems were the independent studies of Neil Gordon, David Salmon and Adrian Smith (bootstrap filter), Genshiro Kitagawa (Monte Carlo filter) , and the one by Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut published in the 1990s. The term interacting "particle filters" was first coined in 1996 by Del Moral. Particle filters were also developed in signal processing in the early 1989-1992 by P. Del Moral, J.C. Noyer, G. Rigal, and G. Salut in the LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on RADAR/SONAR and GPS signal processing problems. The foundations and the first rigorous analysis on the convergence of genetic type models and mean field Feynman-Kac particle methods are due to Pierre Del Moral in 1996. Branching type particle methods with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons, and by Dan Crisan, Pierre Del Moral and Terry Lyons. The first uniform convergence results with respect to the time parameter for mean field particle models were developed in the end of the 1990s by Pierre Del Moral and Alice Guionnet for interacting jump type processes, and by Florent Malrieu for nonlinear diffusion type processes. New classes of mean field particle simulation techniques for Feynman-Kac path-integration problems includes genealogical tree based models, backward particle models, adaptive mean field particle models, island type particle models, and particle Markov chain Monte Carlo methods
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