mean-field particle methods

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.{{cite journal|last1=McKean|first1=Henry, P.|title=Propagation of chaos for a class of non-linear parabolic equations|journal=Lecture Series in Differential Equations, Catholic Univ.|date=1967|volume=7|pages=41–57}}{{cite journal|last1=McKean|first1=Henry, P.|title=A class of Markov processes associated with nonlinear parabolic equations|journal=Proc. Natl. Acad. Sci. USA|date=1966|volume=56|issue=6|pages=1907–1911|doi=10.1073/pnas.56.6.1907|pmid=16591437|pmc=220210|bibcode = 1966PNAS...56.1907M |doi-access=free}} 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,{{cite journal|last1=Braun|first1=Werner|last2=Hepp|first2=Klaus|title=The Vlasov dynamics and its fluctuations in the 1 limit of interacting classical particles.|journal=Communications in Mathematical Physics|date=1977|volume=56|issue=2|pages=101–113|doi=10.1007/bf01611497|bibcode = 1977CMaPh..56..101B |s2cid=55238868|url=http://projecteuclid.org/euclid.cmp/1103901139}} Karl Oelschläger,{{cite journal |title=A martingale approach to the law of large numbers for weakly interacting stochastic processes|first=Karl |last=Oelschläger|journal=Ann. Probab.|year=1984 |volume=12 |issue=2 |pages=458–479 |doi=10.1214/aop/1176993301|doi-access=free }}{{cite journal |title=On the derivation of reaction-diffusion equations as limit of dynamics of systems of moderately interacting stochastic processes|first=Karl |last=Oelschläger|journal=Prob. Th. Rel. Fields |year=1989 |volume=82 |issue=4 |pages=565–586|doi=10.1007/BF00341284 |s2cid=115773110 |doi-access=free }}{{cite journal |title=Large systems of interacting particles and porous medium equation|first=Karl |last=Oelschläger|journal=J. Differential Equations |year=1990 |volume=88 |issue=2 |pages=294–346 |doi=10.1016/0022-0396(90)90101-t|bibcode = 1990JDE....88..294O |doi-access=free }} Gérard Ben Arous and Marc Brunaud,{{cite journal|last1=Ben Arous|first1=Gérard|last2=Brunaud|first2=Marc|title=Méthode de Laplace: Etude variationnelle des fluctuations de diffusions de type "champ moyen"|journal=Stochastics |date=1990|volume=31|pages=79–144|doi=10.1080/03610919008833649}} Donald Dawson, Jean Vaillancourt{{cite journal|last1=Dawson|first1=Donald|title=Stochastic McKean-Vlasov equations|journal=Nonlinear Differential Equations and Applications |last2=Vaillancourt|first2=Jean|date=1995|volume=2|issue=2|pages=199–229|doi=10.1007/bf01295311|s2cid=121652411}} and Jürgen Gärtner,{{cite journal| last1=Dawson |first1=Donald|last2=Gartner|first2=Jurgen|title=Large deviations from the McKean-Vlasov limit for weakly interacting diffusions|journal=Stochastics|date=1987|volume=20|issue=4|pages=247–308|doi=10.1080/17442508708833446|s2cid=122536900}}{{cite journal|last1=Gartner|first1=Jurgen|title=J. GÄRTNER, On the McKean-Vlasov limit for interacting diffusions|journal=Math. Nachr.|date=1988|volume=137|pages=197–248|doi=10.1002/mana.19881370116}} Christian Léonard,{{cite journal|last1=Léonard|first1=Christian|title=Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés|journal=Annales de l'Institut Henri Poincaré|date=1986|volume=22|pages=237–262}} Sylvie Méléard, Sylvie Roelly,{{cite journal|last1=Méléard|first1=Sylvie|author1-link=Sylvie Méléard|last2=Roelly|first2=Sylvie|author2-link=Sylvie Roelly|title=A propagation of chaos result for a system of particles with moderate interaction|journal=Stoch. Proc. And Appl.|date=1987|volume=26|pages=317–332|doi=10.1016/0304-4149(87)90184-0|doi-access=free}} Alain-Sol Sznitman{{cite book|last1=Sznitman|first1=Alain-Sol|author1-link=Alain-Sol Sznitman|title=Topics in propagation of chaos|date=1991|publisher=Springer, Berlin|pages=164–251|quote=Saint-Flour Probability Summer School, 1989}}{{cite journal|last1=Sznitman|first1=Alain-Sol|title=Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated|journal=J. Funct. Anal.|date=1984|volume=36|issue=3|pages=311–336|doi=10.1016/0022-1236(84)90080-6|doi-access=}} and Hiroshi Tanaka{{cite book|last1=Tanaka|first1=Hiroshi|chapter=Limit Theorems for Certain Diffusion Processes with Interaction |series=North-Holland Mathematical Library |title=Stochastic Analysis, Proceedings of the Taniguchi International Symposium on Stochastic Analysis|date=1984|volume=32 |pages=469–488|doi=10.1016/S0924-6509(08)70405-7|isbn=978-0-444-87588-4 }} for diffusion type models; F. Alberto Grünbaum,{{cite journal|last1=Grunbaum.|first1=F. Alberto|title=Propagation of chaos for the Boltzmann equation|journal=Archive for Rational Mechanics and Analysis|date=1971|volume=42|issue=5|pages=323–345|bibcode = 1971ArRMA..42..323G |doi = 10.1007/BF00250440 |s2cid=118165282}} Tokuzo Shiga, Hiroshi Tanaka,{{cite journal|last1=Shiga|first1=Tokuzo|last2=Tanaka|first2=Hiroshi|title=Central limit theorem for a system of Markovian particles with mean field interactions|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete|date=1985|volume=69|issue=3|pages=439–459|doi=10.1007/BF00532743|s2cid=121905550|doi-access=free}}

Sylvie Méléard and Carl Graham{{cite journal|last1=Graham|first1=Carl|title=Non linear diffusions with jumps|journal=Ann. I.H.P.|volume=28|issue=3|pages=393–402|year=1992}}{{cite book |contribution=Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models|title=Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995)|first=Sylvie |last=Méléard|author-link=Sylvie Méléard |year=1996 |volume=1627 |pages=42–95 |doi=10.1007/bfb0093177|series=Lecture Notes in Mathematics |isbn=978-3-540-61397-8 }}{{cite journal|last1=Graham|first1=Carl|last2=Méléard|first2=Sylvie|author2-link=Sylvie Méléard|title=Stochastic particle approximations for generalized Boltzmann models and convergence estimates.|journal=Annals of Probability|date=1997|volume=25|issue=1|pages=115–132|doi=10.1214/aop/1024404281|doi-access=free}} 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.{{cite journal|last1=Herman|first1=Kahn|last2=Harris|first2=Theodore, E.|title=Estimation of particle transmission by random sampling| journal=Natl. Bur. Stand. Appl. Math. Ser.|date=1951|volume=12|pages=27–30| url=https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf}} 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{{cite journal|last1=Turing|first1=Alan M.|title=Computing machinery and intelligence|journal=Mind|volume=LIX|issue=238|pages=433–460|doi=10.1093/mind/LIX.236.433|date=October 1950}}

and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey.{{cite journal|last=Barricelli|first=Nils Aall|year=1954|author-link=Nils Aall Barricelli|title=Esempi numerici di processi di evoluzione|journal=Methodos|pages=45–68}}{{cite journal|last=Barricelli|first=Nils Aall|year=1957|author-link=Nils Aall Barricelli|title=Symbiogenetic evolution processes realized by artificial methods|journal=Methodos|pages=143–182}} The Australian geneticist Alex Fraser also published in 1957 a series of papers on the genetic type simulation of artificial selection of organisms.{{cite journal|last=Fraser|first=Alex|author-link=Alex Fraser (scientist)|year=1957|title=Simulation of genetic systems by automatic digital computers. I. Introduction|journal=Aust. J. Biol. Sci.|volume=10|pages=484–491|doi=10.1071/BI9570484|doi-access=free}}

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.{{cite journal|last1=Del Moral|first1=Pierre|title=Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups|journal=ESAIM Probability & Statistics|date=2003|volume=7|pages=171–208|url=http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a|doi=10.1051/ps:2003001|doi-access=free}}{{cite journal|last1=Assaraf|first1=Roland|last2=Caffarel|first2=Michel|last3=Khelif|first3=Anatole|title=Diffusion Monte Carlo Methods with a fixed number of walkers|journal=Phys. Rev. E|url=http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|date=2000|volume=61|issue=4|pages=4566–4575|doi=10.1103/physreve.61.4566|pmid=11088257|bibcode=2000PhRvE..61.4566A|url-status=dead|archive-url=https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf|archive-date=2014-11-07}}{{cite journal|last1=Caffarel|first1=Michel|last2=Ceperley|first2=David|last3=Kalos|first3=Malvin|title=Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms|journal=Phys. Rev. Lett.|date=1993|volume=71|issue=13|doi=10.1103/physrevlett.71.2159 |bibcode = 1993PhRvL..71.2159C|pages=2159|pmid=10054598}}{{cite journal|last1=Hetherington|first1=Jack, H.|title=Observations on the statistical iteration of matrices|journal=Phys. Rev. A| date=1984|volume=30|issue=2713|doi=10.1103/PhysRevA.30.2713|pages=2713–2719|bibcode = 1984PhRvA..30.2713H }} 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,{{cite journal|last1=Fermi|first1=Enrique|last2=Richtmyer|first2=Robert, D.|title=Note on census-taking in Monte Carlo calculations|journal=LAM|date=1948|volume=805|issue=A|url=http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf|quote=Declassified report Los Alamos Archive}} 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.{{cite journal|last1=Rosenbluth|first1=Marshall, N.|last2=Rosenbluth|first2=Arianna, W.|title=Monte-Carlo calculations of the average extension of macromolecular chains|journal=J. Chem. Phys.|date=1955|volume=23|issue=2|pages=356–359|bibcode = 1955JChPh..23..356R |doi = 10.1063/1.1741967 |s2cid=89611599|doi-access=free}}

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),{{cite journal

| last1 = Gordon | first1 = N. J.

| last2 = Salmond | first2 = D. J. | last3 = Smith | first3 = A. F. M.

| year = 1993

| title = Novel approach to nonlinear/non-Gaussian Bayesian state estimation

| journal = IEE Proceedings F - Radar and Signal Processing

| volume = 140

| issue = 2

| pages = 107–113

| url = https://ieeexplore.ieee.org/document/210672

| archive-url = https://web.archive.org/web/20160905002857/http://ieeexplore.ieee.org/document/210672/?

| url-status = dead

| archive-date = September 5, 2016

| access-date = 2009-09-19

| doi = 10.1049/ip-f-2.1993.0015

}}

Genshiro Kitagawa (Monte Carlo filter)

,{{cite journal

| last = Kitagawa | first = G.

| year = 1996

| title =Monte carlo filter and smoother for non-Gaussian nonlinear state space models

| volume = 5

| issue = 1

| journal = Journal of Computational and Graphical Statistics

| pages = 1–25

| doi = 10.2307/1390750

| jstor = 1390750

}}

and the one by Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut{{cite journal|last1=Carvalho|first1=Himilcon|last2=Del Moral|first2=Pierre|last3=Monin|first3=André|last4=Salut|first4=Gérard|title=Optimal Non-linear Filtering in GPS/INS Integration.|journal=IEEE Transactions on Aerospace and Electronic Systems|date=July 1997|volume=33|issue=3|pages=835|url=http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|bibcode=1997ITAES..33..835C|doi=10.1109/7.599254|s2cid=27966240|access-date=2014-09-03|archive-date=2022-11-10|archive-url=https://web.archive.org/web/20221110053359/https://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf|url-status=dead}} published in the 1990s. The term interacting "particle filters" was first coined in 1996 by Del Moral.{{cite journal|last1 = Del Moral|first1 = Pierre|title = Non Linear Filtering: Interacting Particle Solution.|journal = Markov Processes and Related Fields|date = 1996|volume = 2|issue = 4|pages = 555–580|url = http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf|access-date = 2014-08-29|archive-date = 2016-03-04|archive-url = https://web.archive.org/web/20160304052857/http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf|url-status = dead}} 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.P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : An unified framework for particle solutions

LAAS-CNRS, Toulouse, Research Report no. 91137, DRET-DIGILOG- LAAS/CNRS contract, April (1991).P. Del Moral, G. Rigal, and G. Salut. Nonlinear and non Gaussian particle filters applied to inertial platform repositioning.

LAAS-CNRS, Toulouse, Research Report no. 92207, STCAN/DIGILOG-LAAS/CNRS Convention STCAN no. A.91.77.013, (94p.) September (1991).P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Experimental results.

Convention DRET no. 89.34.553.00.470.75.01, Research report no.2 (54p.), January (1992).P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation. Theoretical results

Convention DRET no. 89.34.553.00.470.75.01, Research report no.3 (123p.), October (1992).P. Del Moral, J.-Ch. Noyer, G. Rigal, and G. Salut. Particle filters in radar signal processing : detection, estimation and air targets recognition.

LAAS-CNRS, Toulouse, Research report no. 92495, December (1992).P. Del Moral, G. Rigal, and G. Salut. Estimation and nonlinear optimal control : Particle resolution in filtering and estimation.

Studies on: Filtering, optimal control, and maximum likelihood estimation. Convention DRET no. 89.34.553.00.470.75.01. Research report no.4 (210p.), January (1993).

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{{cite journal|last1=Del Moral|first1=Pierre|title=Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems|journal=Annals of Applied Probability|date=1998|edition=Publications du Laboratoire de Statistique et Probabilités, 96-15 (1996)|volume=8|issue=2|pages=438–495| url=http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535|doi=10.1214/aoap/1028903535|doi-access=free}} 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,{{cite journal| last1=Crisan|first1=Dan|last2=Gaines |first2=Jessica|last3=Lyons|first3=Terry|title=Convergence of a branching particle method to the solution of the Zakai|journal=SIAM Journal on Applied Mathematics|date=1998|volume=58|issue=5|pages=1568–1590|doi=10.1137/s0036139996307371|s2cid=39982562}}{{cite journal| last1=Crisan| first1=Dan| last2=Lyons|first2=Terry|title=Nonlinear filtering and measure-valued processes|journal=Probability Theory and Related Fields| date=1997 |volume=109 |issue=2| pages=217–244|doi=10.1007/s004400050131| s2cid=119809371|doi-access=free}}{{cite journal|last1=Crisan|first1=Dan|last2=Lyons|first2=Terry|title=A particle approximation of the solution of the Kushner–Stratonovitch equation|journal=Probability Theory and Related Fields|date=1999|volume=115|issue=4|pages=549–578|doi=10.1007/s004400050249|s2cid=117725141|doi-access=free}} and by Dan Crisan, Pierre Del Moral and Terry Lyons.{{cite journal|last1=Crisan|first1=Dan|last2=Del Moral|first2=Pierre|last3=Lyons|first3=Terry|title=Discrete filtering using branching and interacting particle systems|journal=Markov Processes and Related Fields|date=1999|volume=5|issue=3|pages=293–318|url=http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}} 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{{cite journal|last1=Del Moral|first1=Pierre|last2=Guionnet|first2=Alice|title=On the stability of interacting processes with applications to filtering and genetic algorithms|journal=Annales de l'Institut Henri Poincaré|date=2001|volume=37|issue=2|pages=155–194|url=http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps|doi=10.1016/s0246-0203(00)01064-5|bibcode = 2001AIHPB..37..155D }}{{cite journal|last1=Del Moral|first1=Pierre|last2=Guionnet|first2=Alice|title=On the stability of Measure Valued Processes with Applications to filtering|journal=C. R. Acad. Sci. Paris|date=1999|volume=39|issue=1|pages=429–434}} for interacting jump type processes, and by Florent Malrieu for nonlinear diffusion type processes.{{cite journal|last1=Malrieu|first1=Florent|title=Logarithmic Sobolev inequalities for some nonlinear PDE's|journal=Stochastic Process. Appl.|date=2001|volume=95|issue=1|pages=109–132|doi=10.1016/s0304-4149(01)00095-3|s2cid=13915974|doi-access=}}

New classes of mean field particle simulation techniques for Feynman-Kac path-integration problems includes genealogical tree based models,{{cite journal|last1=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=Genealogies and Increasing Propagations of Chaos for Feynman-Kac and Genetic Models|journal=Annals of Applied Probability|date=2001|volume=11|issue=4|pages=1166–1198|url=http://web.maths.unsw.edu.au/~peterdel-moral/spc.ps}} backward particle models,{{cite journal|last1=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|last3=Singh|first3=Sumeetpal, S.|title=A Backward Particle Interpretation of Feynman-Kac Formulae |journal=M2AN|date=2010|volume=44|issue=5|pages=947–976|url=http://hal.inria.fr/docs/00/42/13/56/PDF/RR-7019.pdf|doi=10.1051/m2an/2010048|arxiv=0908.2556|s2cid=14758161}} adaptive mean field particle models,{{cite journal|last1=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|last3=Jasra|first3=Ajay|title=On Adaptive Resampling Procedures for Sequential Monte Carlo Methods|journal=Bernoulli|date=2012|volume=18|issue=1|pages=252–278|url=http://hal.inria.fr/docs/00/33/25/83/PDF/RR-6700.pdf|doi=10.3150/10-bej335|arxiv=1203.0464|s2cid=4506682}} island type particle models,{{cite journal|last1=Vergé|first1=Christelle|last2=Dubarry|first2=Cyrille|last3=Del Moral|first3=Pierre|last4=Moulines|first4=Eric|title=On parallel implementation of Sequential Monte Carlo methods: the island particle model|journal=Statistics and Computing|date=2013|doi=10.1007/s11222-013-9429-x|volume=25|issue=2|pages=243–260|arxiv=1306.3911|bibcode=2013arXiv1306.3911V|s2cid=39379264}}{{cite arXiv|last1=Chopin|first1=Nicolas|last2=Jacob|first2=Pierre, E.|last3=Papaspiliopoulos|first3=Omiros|title=SMC^2: an efficient algorithm for sequential analysis of state-space models|eprint=1101.1528v3|class=stat.CO|year=2011}} and particle Markov chain Monte Carlo methods{{cite journal| last1=Andrieu| first1=Christophe|last2=Doucet|first2=Arnaud|last3=Holenstein|first3=Roman|title=Particle Markov chain Monte Carlo methods|journal=Journal of the Royal Statistical Society, Series B| date=2010|volume=72|issue=3|pages=269–342|doi=10.1111/j.1467-9868.2009.00736.x|doi-access=free}}{{cite arXiv|last1=Del Moral| first1=Pierre |last2=Patras |first2=Frédéric| last3=Kohn|first3=Robert|title=On Feynman-Kac and particle Markov chain Monte Carlo models|eprint=1404.5733|date=2014| class=math.PR }}

In physics, and more particularly in statistical mechanics, these nonlinear evolution equations are often used to describe the statistical behavior of microscopic interacting particles in a fluid or in some condensed matter. In this context, the random evolution of a virtual fluid or a gas particle is represented by McKean-Vlasov diffusion processes, reaction–diffusion systems, or Boltzmann type collision processes.{{cite book|last1=Cercignani|first1=Carlo|last2=Illner|first2= Reinhard|last3=Pulvirenti|first3= Mario|title=The Mathematical Theory of Dilute Gases.|publisher=Springer|date=1994}} As its name indicates, the mean field particle model represents the collective behavior of microscopic particles weakly interacting with their occupation measures. The macroscopic behavior of these many-body particle systems is encapsulated in the limiting model obtained when the size of the population tends to infinity. Boltzmann equations represent the macroscopic evolution of colliding particles in rarefied gases, while McKean Vlasov diffusions represent the macroscopic behavior of fluid particles and granular gases.

In computational physics and more specifically in quantum mechanics, the ground state energies of quantum systems is associated with the top of the spectrum of Schrödinger's operators. The Schrödinger equation is the quantum mechanics version of the Newton's second law of motion of classical mechanics (the mass times the acceleration is the sum of the forces). This equation represents the wave function (a.k.a. the quantum state) evolution of some physical system, including molecular, atomic of subatomic systems, as well as macroscopic systems like the universe.{{cite journal|last1=Schrodinger|first1=Erwin|title=An Undulatory Theory of the Mechanics of Atoms and Molecules|journal=Physical Review| date=1926| volume=28|issue=6|pages=1049–1070|doi=10.1103/physrev.28.1049|bibcode = 1926PhRv...28.1049S }} The solution of the imaginary time Schrödinger equation (a.k.a. the heat equation) is given by a Feynman-Kac distribution associated with a free evolution Markov process (often represented by Brownian motions) in the set of electronic or macromolecular configurations and some potential energy function. The long time behavior of these nonlinear semigroups is related to top eigenvalues and ground state energies of Schrödinger's operators. The genetic type mean field interpretation of these Feynman-Kac models are termed Resample Monte Carlo, or Diffusion Monte Carlo methods. These branching type evolutionary algorithms are based on mutation and selection transitions. During the mutation transition, the walkers evolve randomly and independently in a potential energy landscape on particle configurations. The mean field selection process (a.k.a. quantum teleportation, population reconfiguration, resampled transition) is associated with a fitness function that reflects the particle absorption in an energy well. Configurations with low relative energy are more likely to duplicate. In molecular chemistry, and statistical physics Mean field particle methods are also used to sample Boltzmann-Gibbs measures associated with some cooling schedule, and to compute their normalizing constants (a.k.a. free energies, or partition functions).{{cite journal| last1=Lelièvre|first1=Tony|last2=Rousset|first2=Mathias|last3=Stoltz|first3=Gabriel|title=Computation of free energy differences through nonequilibrium stochastic dynamics: the reaction coordinate case. |journal=J. Comput. Phys.|date=2007|volume=222|issue=2|pages=624–643|doi=10.1016/j.jcp.2006.08.003|arxiv = cond-mat/0603426 |bibcode = 2007JCoPh.222..624L |s2cid=27265236}}{{cite journal|last1=Lelièvre|first1=Tony|last2=Rousset|first2=Mathias|last3=Stoltz|first3=Gabriel|title=Free energy computations: A mathematical perspective|journal=Imperial College Press|date=2010|pages=472}}

In computational biology, and more specifically in population genetics, spatial branching processes with competitive selection and migration mechanisms can also be represented by mean field genetic type population dynamics models.{{cite book|last1=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=Séminaire de Probabilités XXXIV |contribution=Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering |series=Lecture Notes in Mathematics|date=2000|volume=1729|pages=1–145|url=http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf|doi=10.1007/bfb0103798|isbn=978-3-540-67314-9}}{{cite journal|last1=Caron|first1=F.|last2=Del Moral|first2=P.|last3=Pace|first3=M.|last4=Vo|first4=B.-N.|title=On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering|journal=Stochastic Analysis and Applications|volume=29|issue=6|year=2011|pages=951–997|issn=0736-2994|doi=10.1080/07362994.2011.598797|arxiv=1009.1845|s2cid=303252}}

The first moments of the occupation measures of a spatial branching process are given by Feynman-Kac distribution flows.{{cite book|last1=Dynkin|first1=Eugène, B.|title=An Introduction to Branching Measure-Valued Processes|date=1994|publisher=CRM Monograph Series|isbn=978-0-8218-0269-4|pages=134}}{{cite journal|last1=Zoia|first1=Andrea|last2=Dumonteil|first2=Eric|last3=Mazzolo|first3=Alain|title=Discrete Feynman-Kac formulas for branching random walks|journal=EPL|volume=98|issue=40012|pages=40012|doi=10.1209/0295-5075/98/40012|arxiv = 1202.2811 |bibcode = 2012EL.....9840012Z |year=2012|s2cid=119125770}} The mean field genetic type approximation of these flows offers a fixed population size interpretation of these branching processes. Extinction probabilities can be interpreted as absorption probabilities of some Markov process evolving in some absorbing environment. These absorption models are represented by Feynman-Kac models.{{cite journal|last1=Pitman|first1=Jim|last2=Fitzsimmons|first2=Patrick, J.|title=Kac's moment formula and the Feynman–Kac formula for additive functionals of a Markov process|journal=Stochastic Processes and Their Applications|date=1999|volume=79|issue=1|pages=117–134|doi=10.1016/S0304-4149(98)00081-7|url=http://math.ucsd.edu/~pfitz/downloads/kac/kac.html|doi-access=free}}{{cite journal|last1=Arendt|first1=Wolfgang|last2=Batty|first2=Charles, J.K.|title=Absorption semigroups and Dirichlet boundary conditions|journal=Math. Ann.|date=1993|volume=295|pages=427–448|url=https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/downloads/pubbib/short/1993-AreBat-AbsSmgDrcBndCnd.pdf|doi=10.1007/bf01444895|s2cid=14021993}}{{cite journal| last1=Lant|first1=Timothy|last2=Thieme|first2=Horst|title=Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality| journal=Positivity|date=2007|volume=11|issue=2|pages=299–318|doi=10.1007/s11117-006-2044-8|s2cid=54520042}}{{cite journal|last1=Takeda|first1=Masayoshi|title=Some Topics connected with Gaugeability for Feynman-Kac Functionals|journal=RIMS Kokyuroku Bessatsu|date=2008|volume=B6|pages=221–236|url=http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B6/pdf/B6-18.pdf}} The long time behavior of these processes conditioned on non-extinction can be expressed in an equivalent way by quasi-invariant measures, Yaglom limits,{{cite journal|last1=Yaglom|first1=Isaak|title=Certain limit theorems of the theory of branching processes|journal=Dokl. Akad. Nauk SSSR|date=1947|volume=56|pages=795–798}} or invariant measures of nonlinear normalized Feynman-Kac flows.{{cite journal|last1=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|title=Particle Motions in Absorbing Medium with Hard and Soft Obstacles|journal=Stochastic Analysis and Applications|date=2004|volume=22|issue=5|pages=1175–1207|doi=10.1081/SAP-200026444|s2cid=4494495|url=http://web.maths.unsw.edu.au/~peterdel-moral/obstacle.ps}}{{cite journal|last1=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=On the Stability of Non Linear Semigroup of Feynman-Kac Type|journal=Annales de la Faculté des Sciences de Toulouse|date=2002|volume=11|issue=2|pages=135–175| url=http://archive.numdam.org/ARCHIVE/AFST/AFST_2002_6_11_2/AFST_2002_6_11_2_135_0/AFST_2002_6_11_2_135_0.pdf|doi=10.5802/afst.1021}}

In computer sciences, and more particularly in artificial intelligence these mean field type genetic algorithms are used as random search heuristics that mimic the process of evolution to generate useful solutions to complex optimization problems.{{cite book|last1=Kallel|first1=Leila|last2=Naudts|first2=Bart|last3=Rogers|first3=Alex|title=Theoretical Aspects of Evolutionary Computing|publisher=Springer, Berlin, New York; Natural computing series.|isbn=978-3540673965|pages=497|date=2001-05-08}}{{cite journal|last1=Del Moral| first1=Pierre|last2=Kallel|first2=Leila|last3=Rowe|first3=John|title=Modeling genetic algorithms with interacting particle systems|journal=Revista de Matemática: Teoría y Aplicaciones|date=2001|volume=8|issue=2|pages=19–77|citeseerx = 10.1.1.87.7330|doi=10.15517/rmta.v8i2.201}}{{cite journal|last1=Del Moral|first1=Pierre|last2=Guionnet|first2=Alice|title=On the stability of interacting processes with applications to filtering and genetic algorithms|journal=Annales de l'Institut Henri Poincaré|date=2001|volume=37|issue=2|pages=155–194|doi=10.1016/S0246-0203(00)01064-5|bibcode = 2001AIHPB..37..155D }} These stochastic search algorithms belongs to the class of Evolutionary models. The idea is to propagate a population of feasible candidate solutions using mutation and selection mechanisms. The mean field interaction between the individuals is encapsulated in the selection and the cross-over mechanisms.

In mean field games and multi-agent interacting systems theories, mean field particle processes are used to represent the collective behavior of complex systems with interacting individuals.{{cite journal|last=Aumann|first=Robert John|title=Markets with a continuum of traders| journal=Econometrica|date=1964|volume=32|issue=1–2|pages=39–50|doi=10.2307/1913732|jstor=1913732}}{{cite journal|last1=Jovanovic|first1=Boyan|last2=Rosenthal|first2=Robert W.|title=Anonymous sequential games|journal= Journal of Mathematical Economics|date=1988|volume=17|issue=1|pages=77–87|doi=10.1016/0304-4068(88)90029-8}}{{cite journal|last1=Huang|first1=Minyi.Y|last2=Malhame|first2=Roland P.|last3=Caines|first3= Peter E.|title=Large Population Stochastic Dynamic Games: Closed-Loop McKean–Vlasov Systems and the Nash Certainty Equivalence Principle|journal=Communications in Information and Systems|date=2006|volume=6|issue=3|pages=221–252|doi=10.4310/CIS.2006.v6.n3.a5|doi-access=free}}{{cite book|last=Maynard Smith|first=John|title=Evolution and the Theory of Games|publisher=Cambridge University Press, Cambridge|date=1982}}{{cite arXiv|last1=Kolokoltsov|first1=Vassili|last2=Li|first2=Jiajie|last3=Yang|first3=Wei|title=Mean field games and nonlinear Markov processes|eprint=1112.3744v2|class=math.PR|year=2011}}{{cite journal|last1=Lasry|first1=Jean Michel|last2=Lions|first2=Pierre Louis|title=Mean field games|journal=Japanese J. Math|date=2007|volume=2|issue=1|pages=229–260|doi=10.1007/s11537-007-0657-8|s2cid=1963678|url=https://basepub.dauphine.fr/handle/123456789/2263}}{{cite journal|first1=René|last1=Carmona|first2=Jean Pierre|last2=Fouque|first3= Li-Hsien|last3=Sun|title= Mean Field Games and Systemic Risk|journal=Communications in Mathematical Sciences|date=2014|bibcode=2013arXiv1308.2172C|arxiv=1308.2172}}{{cite journal|last1=Budhiraja|first1=Amarjit|last2=Del Moral|first2=Pierre|last3=Rubenthaler|first3=Sylvain|title=Discrete time Markovian agents interacting through a potential|journal=ESAIM Probability & Statistics|date=2013|volume=17|pages=614–634|doi=10.1051/ps/2012014|arxiv=1106.3306|s2cid=28058111}} In this context, the mean field interaction is encapsulated in the decision process of interacting agents. The limiting model as the number of agents tends to infinity is sometimes called the continuum model of agents{{cite journal|last1=Aumann|first1=Robert|title=Markets with a continuum of traders|journal=Econometrica|date=1964|volume=32|issue=1–2|pages=39–50|url=http://www.u.arizona.edu/~mwalker/501BReadings/Aumann1964.pdf|doi=10.2307/1913732|jstor=1913732}}

In information theory, and more specifically in statistical machine learning and signal processing, mean field particle methods are used to sample sequentially from the conditional distributions of some random process with respect to a sequence of observations or a cascade of rare events.{{cite book|last=Del Moral|first=Pierre|title=Feynman-Kac formulae. Genealogical and interacting particle approximations|year=2004 |publisher=Springer|quote=Series: Probability and Applications|url=https://www.springer.com/mathematics/probability/book/978-0-387-20268-6|pages=575|isbn=9780387202686|series=Probability and its Applications}}{{cite book|last1=Del Moral|first1=Pierre|last2=Lézaud|first2=Pascal|title=Branching and interacting particle interpretation of rare event probabilities.|date=2006|publisher=Springer, Berlin|pages=277–323|edition=stochastic Hybrid Systems: Theory and Safety Critical Applications, eds. H. Blom and J. Lygeros.|url=http://web.maths.unsw.edu.au/~peterdel-moral/Del-Moral-Lezaud-rare-events.pdf}} In discrete time nonlinear filtering problems, the conditional distributions of the random states of a signal given partial and noisy observations satisfy a nonlinear updating-prediction evolution equation. The updating step is given by Bayes' rule, and the prediction step is a Chapman-Kolmogorov transport equation. The mean field particle interpretation of these nonlinear filtering equations is a genetic type selection-mutation particle algorithm{{cite journal|last1=Del Moral|first1=Pierre|title=Non Linear Filtering: Interacting Particle Solution.|journal=Markov Processes and Related Fields|date=1996|volume=2|issue=4|pages=555–580|url=http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf|access-date=2014-08-29|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304052857/http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf|url-status=dead}}

During the mutation step, the particles evolve independently of one another according to the Markov transitions of the signal . During the selection stage, particles with small relative likelihood values are killed, while the ones with high relative values are multiplied.{{cite journal|last1=Crisan|first1=Dan|last2=Del Moral|first2=Pierre|last3=Lyons|first3=Terry|title=Discrete Filtering Using Branching and Interacting Particle Systems.|journal=Markov Processes and Related Fields|date=1998|volume=5|issue=3|pages=293–318|url=http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf}}{{cite journal|last1=Crisan|first1=Dan|last2=Del Moral|first2=Pierre|last3=Lyons|first3=Terry|title=Interacting Particle Systems Approximations of the Kushner Stratonovitch Equation|journal=Advances in Applied Probability|date=1998|volume=31|issue=3|pages=819–838|url=http://web.maths.unsw.edu.au/~peterdel-moral/ks-approx.pdf|doi=10.1239/aap/1029955206|hdl=10068/56073|s2cid=121888859 }} These mean field particle techniques are also used to solve multiple-object tracking problems, and more specifically to estimate association measures{{cite journal|last1=Caron|first1=François|last2=Del Moral| first2=Pierre|last3=Doucet|first3=Arnaud|last4=Pace|first4=Michele|title=Particle approximations of a class of branching distribution flows arising in multi-target tracking|journal=SIAM J. Control Optim.|date=2011|volume=49 |issue=4 |pages=1766–1792|url=http://hal.archives-ouvertes.fr/docs/00/46/41/30/PDF/RR-7233.pdf|doi=10.1137/100788987|arxiv=1012.5360|s2cid=6899555}}{{cite journal|last1=Pace|first1=Michele|last2=Del Moral|first2=Pierre|title=Mean-Field PHD Filters Based on Generalized Feynman-Kac Flow|journal=IEEE Journal of Selected Topics in Signal Processing|date=2013|volume=7|issue= 3|pages=484–495|doi=10.1109/JSTSP.2013.2250909|bibcode = 2013ISTSP...7..484P |s2cid=15906417}}

The continuous time version of these particle models are mean field Moran type particle interpretations of the robust optimal filter evolution equations or the Kushner-Stratonotich stochastic partial differential equation.{{cite journal|last1=Del Moral|first1=Pierre|last2=Miclo|first2=Laurent|title=A Moran particle system approximation of Feynman-Kac formulae.|journal=Stochastic Processes and Their Applications|date=2000|volume=86|issue=2|pages=193–216|doi=10.1016/S0304-4149(99)00094-0|doi-access=}} These genetic type mean field particle algorithms also termed Particle Filters and Sequential Monte Carlo methods are extensively and routinely used in operation research and statistical inference

.

{{cite book

| last1 = Cappe | first1 = O.

| last2 = Moulines | first2 = E. | last3 = Ryden | first3 = T.

| year = 2005

| title = Inference in Hidden Markov Models

| publisher = Springer

}}

{{cite book

| last = Liu | first = J.

| year = 2001

| title = Monte Carlo strategies in Scientific Computing

| publisher = Springer

}}

{{cite book

| last1 = Doucet | first1 = A.

| editor1-last = de Freitas | editor1-first = J. F. G. | editor2-first = J. | editor2-last = Gordon

| year = 2001

| title = Sequential Monte Carlo Methods in Practice

| publisher = Springer

}}

The term "particle filters" was first coined in 1996 by Del Moral, and the term "sequential Monte Carlo" by Liu and Chen in 1998. Subset simulation and Monte Carlo splitting{{cite journal|last1=Botev|first1=Z. I. |last2=Kroese|first2=D. P.|author-link2=Dirk Kroese|title=Efficient Monte Carlo simulation via the generalized splitting method|journal=Methodology and Computing in Applied Probability|date=2008|volume=10|issue=4|pages=471–505|doi=10.1007/s11009-008-9073-7|citeseerx=10.1.1.399.7912 |s2cid=1147040 }} techniques are particular instances of genetic particle schemes and Feynman-Kac particle models equipped with Markov chain Monte Carlo mutation transitions{{cite journal|last1=Del Moral|first1=Pierre|last2=Doucet|first2=Arnaud|last3=Jasra|first3=Ajay|title=Sequential Monte Carlo samplers|journal=Journal of the Royal Statistical Society, Series B (Statistical Methodology)|date=2006|volume=68|issue=3|pages=411–436|url=http://web.maths.unsw.edu.au/~peterdel-moral/smc_samplers_try.pdf|doi=10.1111/j.1467-9868.2006.00553.x|arxiv=cond-mat/0212648|s2cid=12074789}}{{cite journal|last1=Botev|first1=Z. I. |last2=Kroese|first2=D. P.|title=Efficient Monte Carlo simulation via the generalized splitting method|journal=Statistics and Computing|date=2012|volume=22|issue=1|pages=1–16|doi=10.1007/s11222-010-9201-4|s2cid=14970946 }}{{cite journal|last1=Cérou|first1=Frédéric|last2=Del Moral|first2=Pierre|last3=Furon|first3=Teddy|last4=Guyader|first4=Arnaud|title=Sequential Monte Carlo for Rare event estimation|journal=Statistics and Computing|date=2012|volume=22|issue=3|pages=795–808|doi=10.1007/s11222-011-9231-6|s2cid=16097360|url=https://hal.inria.fr/inria-00584352/file/cdfg.pdf}}

To motivate the mean field simulation algorithm we start with S a finite or countable state space and let P(S) denote the set of all probability measures on S. Consider a sequence of probability distributions $(\backslash eta\_0,\; \backslash eta\_1,\; \backslash cdots)$ on S satisfying an evolution equation:

{{NumBlk|:| $\backslash eta\_\{n+1\}=\backslash Phi(\backslash eta\_n)$|{{EquationRef|1}}}}

for some, possibly nonlinear, mapping $\backslash Phi:\; P(S)\; \backslash to\; P(S).$ These distributions are given by vectors

:$\backslash eta\_n=(\backslash eta\_n(x))\_\{x\backslash in\; S\},$

that satisfy:

:$0\; \backslash leqslant\; \backslash eta\_n(x)\; \backslash leqslant\; 1,\; \backslash qquad\; \backslash sum\backslash nolimits\_\{x\backslash in\; S\}\backslash eta\_n(x)=1.$

Therefore, $\backslash Phi$ is a mapping from the $(s-1)$-unit simplex into itself, where s stands for the cardinality of the set S. When s is too large, solving equation ({{EquationNote|1}}) is intractable or computationally very costly. One natural way to approximate these evolution equations is to reduce sequentially the state space using a mean field particle model. One of the simplest mean field simulation scheme is defined by the Markov chain

:$\backslash xi^\{(N)\}\_n=\backslash left(\backslash xi^\{(N,1)\}\_n,\; \backslash cdots,\; \backslash xi^\{(N,N)\}\_n\; \backslash right)$

on the product space $S^N$, starting with N independent random variables with probability distribution $\backslash eta\_0$ and elementary transitions

:$\backslash mathbf\{P\}\; \backslash left(\; \backslash left.\; \backslash xi^\{(N,1)\}\_\{n+1\}=y^1,\backslash cdots,\backslash xi^\{(N,N)\}\_\{n+1\}=y^N\; \backslash right\; |\backslash xi^\{(N)\}\_n\backslash right)=\backslash prod\_\{i=1\}^N\; \backslash Phi\backslash left(\backslash eta\_n^N\backslash right)\backslash left(y^i\backslash right),$

with the empirical measure

:$\backslash eta^N\_n=\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N1\_\{\backslash xi^\{(N,j)\}\_n\}$

where $1\_x$ is the indicator function of the state x.

In other words, given $\backslash xi^\{(N)\}\_n$ the samples $\backslash xi^\{(N)\}\_\{n+1\}$ are independent random variables with probability distribution $\backslash Phi\backslash left(\backslash eta\_n^N\backslash right)$. The rationale behind this mean field simulation technique is the following: We expect that when $\backslash eta\_\{n\}^N$ is a good approximation of $\backslash eta\_n$, then $\backslash Phi\backslash left(\backslash eta\_n^N\backslash right)$ is an approximation of $\backslash Phi\backslash left(\backslash eta\_n\backslash right)=\backslash eta\_\{n+1\}$. Thus, since $\backslash eta\_\{n+1\}^N$ is the empirical measure of N conditionally independent random variables with common probability distribution $\backslash Phi\backslash left(\backslash eta\_n^N\backslash right)$, we expect $\backslash eta\_\{n+1\}^N$ to be a good approximation of $\backslash eta\_\{n+1\}$.

Another strategy is to find a collection

:$K\_\{\backslash eta\_n\}=\backslash left(K\_\{\backslash eta\_n\}(x,y)\backslash right)\_\{x,y\backslash in\; S\}$

of stochastic matrices indexed by $\backslash eta\_n\backslash in\; P(S)$ such that

{{NumBlk|:| $\backslash sum\_\{x\backslash in\; S\}\backslash eta\_n(x)K\_\{\backslash eta\_n\}(x,y)=\backslash Phi(\backslash eta\_n)(y)=\backslash eta\_\{n+1\}(y)$|{{EquationRef|2}}}}

This formula allows us to interpret the sequence $(\backslash eta\_0,\; \backslash eta\_1,\; \backslash cdots)$ as the probability distributions of the random states $\backslash left(\backslash overline\{X\}\_0,\; \backslash overline\{X\}\_1,\; \backslash cdots\; \backslash right)$ of the nonlinear Markov chain model with elementary transitions

:$\backslash mathbf\{P\}\; \backslash left\; (\; \backslash left.\backslash overline\{X\}\_\{n+1\}=y\; \backslash right|\; \backslash overline\{X\}\_n=x\; \backslash right\; )=K\_\{\backslash eta\_n\}(x,y),\; \backslash qquad\; \backslash text\{Law\}(\backslash overline\{X\}\_n)=\backslash eta\_n.$

A collection of Markov transitions $K\_\{\backslash eta\_n\}$ satisfying the equation ({{EquationNote|1}}) is called a McKean interpretation of the sequence of measures $\backslash eta\_n$.

The mean field particle interpretation of ({{EquationNote|2}}) is now defined by the Markov chain

:$\backslash xi^\{(N)\}\_n=\backslash left(\backslash xi^\{(N,1)\}\_n,\; \backslash cdots,\; \backslash xi^\{(N,N)\}\_n\; \backslash right)$

on the product space $S^N$, starting with N independent random copies of $X\_0$ and elementary transitions

:$\backslash mathbf\{P\}\backslash left(\; \backslash left.\; \backslash xi^\{(N,1)\}\_\{n+1\}=y^1,\backslash cdots,\backslash xi^\{(N,N)\}\_\{n+1\}=y^N\; \backslash right\; |\backslash xi^\{(N)\}\_n\backslash right)=\backslash prod\_\{i=1\}^N\; K\_\{n+1,\backslash eta\_n^N\}\backslash left(\backslash xi^\{(N,i)\}\_n,y^i\backslash right),$

with the empirical measure

:$\backslash eta^N\_n=\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N1\_\{\backslash xi^\{(N,j)\}\_n\}$

Under some weak regularity conditions on the mapping $\backslash Phi$ for any function $f:\; S\backslash to\; \backslash mathbf\{R\}$, we have the almost sure convergence

:$\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; f\backslash left(\backslash xi^\{(N,j)\}\_n\backslash right)\backslash to\_\{N\backslash uparrow\backslash infty\}E\backslash left(f(\backslash overline\{X\}\_n)\backslash right)=\backslash sum\_\{x\backslash in\; S\}\backslash eta\_n(x)f(x)$

These nonlinear Markov processes and their mean field particle interpretation can be extended to time non homogeneous models on general measurable state spaces.

To illustrate the abstract models presented above, we consider a stochastic matrix $M=(M(x,y))\_\{x,y\backslash in\; S\}$ and some function $G\; :\; S\; \backslash to\; (0,1)$. We associate with these two objects the mapping

:$\backslash begin\{cases\}\; \backslash Phi\; :\; P(S)\; \backslash to\; P(S)\; \backslash \backslash \; (\backslash eta\_n(x))\_\{x\backslash in\; S\}\; \backslash mapsto\; \backslash left(\backslash Phi(\backslash eta\_n)(y)\backslash right)\_\{y\backslash in\; S\}\; \backslash end\{cases\}\; \backslash qquad\; \backslash Phi(\backslash eta\_n)(y)=\backslash sum\_\{x\backslash in\; S\}\; \backslash Psi\_\{G\}(\backslash eta\_n)(x)M(x,y)$

and the Boltzmann-Gibbs measures $\backslash Psi\_\{G\}(\backslash eta\_n)(x)$ defined by

:$\backslash Psi\_\{G\}(\backslash eta\_n)(x)=\backslash frac\{\backslash eta\_n(x)G(x)\}\{\backslash sum\_\{z\backslash in\; S\}\backslash eta\_n(z)G(z)\}.$

We denote by $K\_\{\backslash eta\_n\}=\backslash left(K\_\{\backslash eta\_n\}(x,y)\backslash right)\_\{x,y\backslash in\; S\}$ the collection of stochastic matrices indexed by $\backslash eta\_n\backslash in\; P(S)$ given by

:$K\_\{\backslash eta\_n\}(x,y)=\backslash epsilon\; G(x)\; M(x,y)+(1-\backslash epsilon\; G(x))\; \backslash Phi(\backslash eta\_n)(y)$

for some parameter $\backslash epsilon\; \backslash in\; [0,1]$. It is readily checked that the equation ({{EquationNote|2}}) is satisfied. In addition, we can also show (cf. for instance) that the solution of ({{EquationNote|1}}) is given by the Feynman-Kac formula

:$\backslash eta\_n(x)\; =\backslash frac\{E\backslash left(1\_x(X\_n)\backslash prod\_\{p=0\}^\{n-1\}\; G(X\_p)\backslash right)\}\{E\backslash left(\backslash prod\_\{p=0\}^\{n-1\}\; G(X\_p)\; \backslash right)\},$

with a Markov chain $X\_n$ with initial distribution $\backslash eta\_0$ and Markov transition M.

For any function $f\; :\; S\backslash to\; \backslash mathbf\{R\}$ we have

:$\backslash eta\_n(f):=\backslash sum\_\{x\backslash in\; S\}\backslash eta\_n(x)f(x)\; =\backslash frac\{E\backslash left(f(X\_n)\backslash prod\_\{p=0\}^\{n-1\}G(X\_p)\backslash right)\}\{E\backslash left(\backslash prod\_\{p=0\}^\{n-1\}\; G(X\_p)\backslash right)\}$

If $G(x)=1$ is the unit function and $\backslash epsilon=1$, then we have

:$K\_\{\backslash eta\_n\}(x,y)=M(x,y)=\backslash mathbf\{P\}\; \backslash left(\; \backslash left.\; X\_\{n+1\}=y\; \backslash right\; |\; X\_n=x\backslash right),\; \backslash qquad\; \backslash eta\_n(x)\; =E\backslash left(1\_x(X\_n)\backslash right)=\backslash mathbf\{P\}(X\_n=x).$

And the equation ({{EquationNote|2}}) reduces to the Chapman-Kolmogorov equation

:$\backslash eta\_\{n+1\}(y)=\backslash sum\_\{x\backslash in\; S\}\backslash eta\_n(x)M(x,y)\; \backslash qquad\; \backslash Leftrightarrow\; \backslash qquad\; \backslash mathbf\{P\}\backslash left(X\_\{n+1\}=y\backslash right)\; =\backslash sum\_\{x\backslash in\; S\}\; \backslash mathbf\{P\}(X\_\{n+1\}=y|X\_n=x)\; \backslash mathbf\{P\}\backslash left(X\_n=x\backslash right)$

The mean field particle interpretation of this Feynman-Kac model is defined by sampling sequentially N conditionally independent random variables $\backslash xi^\{(N,i)\}\_\{n+1\}$ with probability distribution

:$K\_\{n+1,\backslash eta\_n^N\}\backslash left(\backslash xi^\{(N,i)\}\_n,y\backslash right)=\backslash epsilon\; G\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)\; M\backslash left(\backslash xi^\{(N,i)\}\_n,y\backslash right)+\backslash left(1-\backslash epsilon\; G\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)\backslash right)\; \backslash sum\_\{j=1\}^N\; \backslash frac\{G\backslash left(\backslash xi^\{(N,j)\}\_n\backslash right)\}\{\backslash sum\_\{k=1\}^N\; G\backslash left(\backslash xi^\{(N,k)\}\_n\backslash right)\}\; M\backslash left(\backslash xi^\{(N,j)\}\_n,y\backslash right)$

In other words, with a probability $\backslash epsilon\; G\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)$ the particle $\backslash xi^\{(N,i)\}\_n$ evolves to a new state $\backslash xi^\{(N,i)\}\_\{n+1\}=y$ randomly chosen with the probability distribution $M\backslash left(\backslash xi^\{(N,i)\}\_n,y\backslash right)$; otherwise, $\backslash xi^\{(N,i)\}\_n$ jumps to a new location $\backslash xi^\{(N,j)\}\_\{n\}$ randomly chosen with a probability proportional to $G\backslash left(\backslash xi^\{(N,j)\}\_n\backslash right)$ and evolves to a new state $\backslash xi^\{(N,i)\}\_\{n+1\}=y$ randomly chosen with the probability distribution $M\backslash left(\backslash xi^\{(N,j)\}\_n,\; y\backslash right).$ If $G(x)=1$ is the unit function and $\backslash epsilon=1$, the interaction between the particle vanishes and the particle model reduces to a sequence of independent copies of the Markov chain $X\_n$. When $\backslash epsilon=0$ the mean field particle model described above reduces to a simple mutation-selection genetic algorithm with fitness function G and mutation transition M. These nonlinear Markov chain models and their mean field particle interpretation can be extended to time non homogeneous models on general measurable state spaces (including transition states, path spaces and random excursion spaces) and continuous time models.

We consider a sequence of real valued random variables $\backslash left\; (\backslash overline\{X\}\_0,\; \backslash overline\{X\}\_1,\; \backslash cdots\; \backslash right)$ defined sequentially by the equations

{{NumBlk|:| $\backslash overline\{X\}\_\{n+1\}=E\backslash left(a\backslash left(\backslash overline\{X\}\_n\backslash right)\backslash right)\; b\; \backslash left\; (\backslash overline\{X\}\_n\; \backslash right\; )+c\; \backslash left\; (\backslash overline\{X\}\_n\; \backslash right\; )+\backslash sigma\; W\_n$|{{EquationRef|3}}}}

with a collection $W\_n$ of independent standard Gaussian random variables, a positive parameter σ, some functions $a,b,c:\; \backslash mathbf\{R\}\; \backslash to\; \backslash mathbf\{R\},$ and some standard Gaussian initial random state $\backslash overline\{X\}\_0$. We let $\backslash eta\_n$ be the probability distribution of the random state $\backslash overline\{X\}\_n$; that is, for any bounded measurable function f, we have

:$E\backslash left(f(\backslash overline\{X\}\_n)\backslash right)=\backslash int\_\{\backslash mathbf\{R\}\}\; f(x)\; \backslash eta\_n(dx),$

with

:$\backslash mathbf\{P\}\; \backslash left\; (\backslash overline\{X\}\_n\backslash in\; dx\; \backslash right\; )=\backslash eta\_n(dx)$

The integral is the Lebesgue integral, and dx stands for an infinitesimal neighborhood of the state x. The Markov transition of the chain is given for any bounded measurable functions f by the formula

:$E\backslash left(\; \backslash left.\; f\; \backslash left\; (\backslash overline\{X\}\_\{n+1\}\; \backslash right\; )\; \backslash right\; |\backslash overline\{X\}\_n=x\backslash right)=\backslash int\_\{\backslash mathbf\{R\}\}\; K\_\{\backslash eta\_n\}(x,dy)\; f(y),$

with

:$K\_\{\backslash eta\_n\}(x,dy)=\backslash mathbf\{P\}\; \backslash left\; (\; \backslash left.\backslash overline\{X\}\_\{n+1\}\backslash in\; dy\backslash right\; |\; \backslash overline\{X\}\_n=x\; \backslash right\; )=\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\backslash sigma\}\; \backslash exp\{\backslash left\backslash \{-\backslash frac\{1\}\{2\backslash sigma^2\}\backslash left(y-\backslash left[b(x)\backslash int\_\{\backslash mathbf\{R\}\}\; a(z)\; \backslash eta\_n(dz)+c(x)\backslash right]\backslash right)^2\backslash right\backslash \}\}\; dy$

Using the tower property of conditional expectations we prove that the probability distributions $\backslash eta\_n$ satisfy the nonlinear equation

:$\backslash int\_\{\backslash mathbf\{R\}\}\; \backslash eta\_\{n+1\}(dy)\; f(y)=\backslash int\_\{\backslash mathbf\{R\}\}\backslash left[\backslash int\_\{\backslash mathbf\{R\}\}\; \backslash eta\_n(dx)K\_\{\backslash eta\_n\}(x,dy)\backslash right]\; f(y)$

for any bounded measurable functions f. This equation is sometimes written in the more synthetic form

:$\backslash eta\_\{n+1\}\; =\backslash Phi\backslash left(\backslash eta\_n\backslash right)=\; \backslash eta\_nK\_\{\backslash eta\_n\}\backslash quad\backslash Leftrightarrow\backslash quad\backslash eta\_\{n+1\}(dy)=\; \backslash left(\backslash eta\_nK\_\{\backslash eta\_n\}\backslash right)(dy)\; =\backslash int\_\{x\backslash in\; \backslash mathbf\{R\}\}\backslash eta\_n(dx)K\_\{\backslash eta\_n\}(x,dy)$

The mean field particle interpretation of this model is defined by the Markov chain

:$\backslash xi^\{(N)\}\_n=\backslash left(\backslash xi^\{(N,1)\}\_n,\; \backslash cdots,\; \backslash xi^\{(N,N)\}\_n\; \backslash right)$

on the product space $\backslash mathbf\{R\}^N$ by

:$\backslash xi^\{(N,i)\}\_\{n+1\}=\backslash left(\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; a\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)\backslash right)\; b\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)+c\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)+\backslash sigma\; W^i\_n\backslash qquad\; 1\backslash leqslant\; i\backslash leqslant\; N$

where

:$\backslash xi^\{(N)\}\_0=\; \backslash left(\backslash xi^\{(N,1)\}\_0,\; \backslash cdots,\; \backslash xi^\{(N,N)\}\_0\backslash right),\; \backslash qquad\; \backslash left(\; W^1\_n,\; \backslash cdots,\; W^N\_n\backslash right)$

stand for N independent copies of $\backslash overline\{X\}\_0$ and $W\_n;\; n\; \backslash geqslant\; 1,$ respectively. For regular models (for instance for bounded Lipschitz functions a, b, c) we have the almost sure convergence

:$\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; f\backslash left(\backslash xi^\{(N,i)\}\_n\backslash right)=\backslash int\_\{\backslash mathbf\{R\}\}\; f(y)\; \backslash eta^N\_n(dy)\; \backslash to\_\{N\backslash uparrow\backslash infty\}\; E\backslash left(f(\backslash overline\{X\}\_n)\backslash right)\; =\; \backslash int\_\{\backslash mathbf\{R\}\}f(y)\backslash eta\_n(dy),$

with the empirical measure

:$\backslash eta^N\_n=\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; \backslash delta\_\{\backslash xi^\{(N,i)\}\_n\}$

for any bounded measurable functions f (cf. for instance ). In the above display, $\backslash delta\_x$ stands for the Dirac measure at the state x.

We consider a standard Brownian motion $\backslash overline\{W\}\_\{t\_n\}$ (a.k.a. Wiener Process) evaluated on a time mesh sequence σ by $b(x)\; \backslash times\; h$ and $\backslash sigma\; \backslash times\; \backslash sqrt\{h\}$, and we write $\backslash overline\{X\}\_\{t\_n\}$ instead of $\backslash overline\{X\}\_n$ the values of the random states evaluated at the time step $t\_n.$ Recalling that $\backslash left(\backslash overline\{W\}\_\{t\_\{n+1\}\}-\backslash overline\{W\}\_\{t\_n\}\backslash right)$ are independent centered Gaussian random variables with variance $t\_n-t\_\{n-1\}\; =\; h,$ the resulting equation can be rewritten in the following form

{{NumBlk|:| $\backslash overline\{X\}\_\{t\_\{n+1\}\}-\backslash overline\{X\}\_\{t\_n\}=E\backslash left(a\backslash left(\backslash overline\{X\}\_\{t\_n\}\backslash right)\backslash right)b(\backslash overline\{X\}\_\{t\_n\})h+\backslash sigma\backslash left(\backslash overline\{W\}\_\{t\_\{n+1\}\}-\backslash overline\{W\}\_\{t\_n\}\backslash right)$|{{EquationRef|4}}}}

When h → 0, the above equation converge to the nonlinear diffusion process

:$d\backslash overline\{X\}\_\{t\}=E\backslash left(a\backslash left(\backslash overline\{X\}\_\{t\}\backslash right)\backslash right)b(\backslash overline\{X\}\_\{t\})dt+\backslash sigma\; d\backslash overline\{W\}\_\{t\}$

The mean field continuous time model associated with these nonlinear diffusions is the (interacting) diffusion process $\backslash xi^\{(N)\}\_t=\backslash left(\backslash xi^\{(N,i)\}\_t\backslash right)\_\{1\backslash leqslant\; i\backslash leqslant\; N\}$ on the product space $\backslash mathbf\{R\}^N$ defined by

:$d\backslash xi^\{(N,i)\}\_\{t\}=\backslash left(\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; a\backslash left(\backslash xi^\{(N,i)\}\_t\backslash right)\backslash right)b\backslash left(\backslash xi^\{(N,i)\}\_t\backslash right)+\backslash sigma\; d\backslash overline\{W\}\_\{t\}^i\backslash qquad\; 1\backslash leqslant\; i\backslash leqslant\; N$

where

:$\backslash xi^\{(N)\}\_0=\; \backslash left(\backslash xi^\{(N,1)\}\_0,\; \backslash cdots,\; \backslash xi^\{(N,N)\}\_0\backslash right),\; \backslash qquad\; \backslash left(\; \backslash overline\{W\}\_\{t\}^1,\; \backslash cdots,\; \backslash overline\{W\}\_t^N\backslash right)$

are N independent copies of $\backslash overline\{X\}\_0$ and $\backslash overline\{W\}\_t.$ For regular models (for instance for bounded Lipschitz functions a, b) we have the almost sure convergence

:$\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; f\backslash left(\backslash xi^\{(N,i)\}\_t\backslash right)=\backslash int\_\{\backslash mathbf\{R\}\}\; f(y)\; \backslash eta^N\_t(dy)\backslash to\_\{N\backslash uparrow\backslash infty\}\; E\backslash left(f(\backslash overline\{X\}\_t)\backslash right)=\backslash int\_\{\backslash mathbf\{R\}\}\; f(y)\; \backslash eta\_t(dy)$,

with $\backslash eta\_t=\backslash text\{Law\}\backslash left(\backslash overline\{X\}\_\{t\}\backslash right),$ and the empirical measure

:$\backslash eta^N\_t=\backslash frac\{1\}\{N\}\backslash sum\_\{j=1\}^N\; \backslash delta\_\{\backslash xi^\{(N,i)\}\_t\}$

for any bounded measurable functions f (cf. for instance.). These nonlinear Markov processes and their mean field particle interpretation can be extended to interacting jump-diffusion processes

{{Reflist}}

{{refbegin}}

• [http://web.maths.unsw.edu.au/~peterdel-moral/simulinks.html Feynman-Kac models and interacting particle systems], theoretical aspects and a list of application domains of Feynman-Kac particle methods
• [http://www.stats.ox.ac.uk/~doucet/smc_resources.html Sequential Monte Carlo method and particle filters resources]
• [http://vserver1.cscs.lsa.umich.edu/~crshalizi/notabene/interacting-particle-systems.html Interacting Particle Systems resources]
• [http://vallico.net/casinoqmc/ QMC in Cambridge and around the world], general information about Quantum Monte Carlo
• EVOLVER Software package for stochastic optimisation using genetic algorithms
• CASINO Quantum Monte Carlo program developed by the Theory of Condensed Matter group at the Cavendish Laboratory in Cambridge
• [https://alea.bordeaux.inria.fr/biips/doku.php Biips is a probabilistic programming software for Bayesian inference with interacting particle systems.]

{{refend}}

{{Stochastic processes}}

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