Your browser does not fully support modern features. Please upgrade for a smoother experience.
Submitted Successfully!
Thank you for your contribution! You can also upload a video entry or images related to this topic. For video creation, please contact our Academic Video Service.
Version Summary Created by Modification Content Size Created at Operation
1 handwiki Vivi Li -- 4512 2022-10-26 01:45:07

Video Upload Options

We provide professional Academic Video Service to translate complex research into visually appealing presentations. Would you like to try it?
No, upload directly Yes
Cite
If you have any further questions, please contact Encyclopedia Editorial Office.
HandWiki. Mean Field Particle Methods. Encyclopedia. Available online: https://encyclopedia.pub/entry/31301 (accessed on 17 October 2026).
HandWiki. Mean Field Particle Methods. Encyclopedia. Available at: https://encyclopedia.pub/entry/31301. Accessed October 17, 2026.
HandWiki. "Mean Field Particle Methods" Encyclopedia, https://encyclopedia.pub/entry/31301 (accessed October 17, 2026).
HandWiki. (2022, October 26). Mean Field Particle Methods. In Encyclopedia. https://encyclopedia.pub/entry/31301
HandWiki. "Mean Field Particle Methods." Encyclopedia. Web. 26 October, 2022.
Mean Field Particle Methods
Edit
The content is sourced from: https://handwiki.org/wiki/Mean_field_particle_methods

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

gas model model mean field

References

  1. McKean, Henry, P. (1967). "Propagation of chaos for a class of non-linear parabolic equations". Lecture Series in Differential Equations, Catholic Univ. 7: 41–57. 
  2. McKean, Henry, P. (1966). "A class of Markov processes associated with nonlinear parabolic equations". Proc. Natl. Acad. Sci. USA 56 (6): 1907–1911. doi:10.1073/pnas.56.6.1907. PMID 16591437. Bibcode: 1966PNAS...56.1907M.  http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=220210
  3. Braun, Werner; Hepp, Klaus (1977). "The Vlasov dynamics and its fluctuations in the 1 limit of interacting classical particles.". Communications in Mathematical Physics 56 (2): 101–113. doi:10.1007/bf01611497. Bibcode: 1977CMaPh..56..101B.  https://dx.doi.org/10.1007%2Fbf01611497
  4. Oelschläger, Karl (1984). "A martingale approach to the law of large numbers for weakly interacting stochastic processes". Ann. Probab. 12 (2): 458–479. doi:10.1214/aop/1176993301.  https://dx.doi.org/10.1214%2Faop%2F1176993301
  5. Oelschläger, Karl (1989). "On the derivation of reaction-diffusion equations as limit of dynamics of systems of moderately interacting stochastic processes". Prob. Th. Rel. Fields 82 (4): 565–586. doi:10.1007/BF00341284.  https://dx.doi.org/10.1007%2FBF00341284
  6. Oelschläger, Karl (1990). "Large systems of interacting particles and porous medium equation". J. Differential Equations 88 (2): 294–346. doi:10.1016/0022-0396(90)90101-t. Bibcode: 1990JDE....88..294O.  https://dx.doi.org/10.1016%2F0022-0396%2890%2990101-t
  7. Ben Arous, Gérard; Brunaud, Marc (1990). "Méthode de Laplace: Etude variationnelle des fluctuations de diffusions de type "champ moyen"". Stochastics 31, 79–144, (1990) 31: 79–144. doi:10.1080/03610919008833649.  https://dx.doi.org/10.1080%2F03610919008833649
  8. Dawson, Donald; Vaillancourt, Jean (1995). "Stochastic McKean-Vlasov equations". Nonlinear Differential Equations and Applications Nodea 2 (2): 199–229. doi:10.1007/bf01295311.  https://dx.doi.org/10.1007%2Fbf01295311
  9. Dawson, Donald; Gartner, Jurgen (1987). "Large deviations from the McKean-Vlasov limit for weakly interacting diffusions". Stochastics 20 (4): 247–308. doi:10.1080/17442508708833446. https://semanticscholar.org/paper/5f0734f46a1f6d3f0a64e9d5b94291d7f9163d58. 
  10. Gartner, Jurgen (1988). "J. GÄRTNER, On the McKean-Vlasov limit for interacting diffusions". Math. Nachr. 137: 197–248. doi:10.1002/mana.19881370116.  https://dx.doi.org/10.1002%2Fmana.19881370116
  11. Léonard, Christian (1986). "Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés". Ann. I.H.P. 22: 237–262. 
  12. Méléard, Sylvie; Roelly, Sylvie (1987). "A propagation of chaos result for a system of particles with moderate interaction". Stoch. Proc. And Appl. 26: 317–332. doi:10.1016/0304-4149(87)90184-0.  https://dx.doi.org/10.1016%2F0304-4149%2887%2990184-0
  13. Sznitman, Alain-Sol (1991). Topics in propagation of chaos. Springer, Berlin. pp. 164–251. "Saint-Flour Probability Summer School, 1989" 
  14. Sznitman, Alain-Sol (1984). "Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated". J. Funct. Anal. 36 (3): 311–336. doi:10.1016/0022-1236(84)90080-6.  https://dx.doi.org/10.1016%2F0022-1236%2884%2990080-6
  15. Tanaka, Hiroshi (1984). "Tanaka, H.: Limit theorems for certain diffusion processes with interaction". Proceedings of the Taniguchi International Symposium on Stochastic Analysis: 469–488. doi:10.1016/S0924-6509(08)70405-7.  https://dx.doi.org/10.1016%2FS0924-6509%2808%2970405-7
  16. Grunbaum., F. Alberto (1971). "Propagation of chaos for the Boltzmann equation". Archive for Rational Mechanics and Analysis 42 (5): 323–345. doi:10.1007/BF00250440. Bibcode: 1971ArRMA..42..323G.  https://dx.doi.org/10.1007%2FBF00250440
  17. Shiga, Tokuzo; Tanaka, Hiroshi (1985). "Central limit theorem for a system of Markovian particles with mean field interactions". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 69 (3): 439–459. doi:10.1007/BF00532743.  https://dx.doi.org/10.1007%2FBF00532743
  18. Graham, Carl (1992). "Non linear diffusions with jumps". Ann. I.H.P. 28 (3): 393–402. 
  19. Méléard, Sylvie (1996). "Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models". Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995). Lecture Notes in Mathematics. 1627. pp. 42–95. doi:10.1007/bfb0093177. ISBN 978-3-540-61397-8.  https://dx.doi.org/10.1007%2Fbfb0093177
  20. Graham, Carl; Méléard, Sylvie (1997). "Stochastic particle approximations for generalized Boltzmann models and convergence estimates.". Annals of Probability 25 (1): 115–132. doi:10.1214/aop/1024404281.  https://dx.doi.org/10.1214%2Faop%2F1024404281
  21. Herman, Kahn; Harris, Theodore, E. (1951). "Estimation of particle transmission by random sampling". Natl. Bur. Stand. Appl. Math. Ser. 12: 27–30. https://dornsifecms.usc.edu/assets/sites/520/docs/kahnharris.pdf. 
  22. Turing, Alan M. (October 1950). "Computing machinery and intelligence". Mind LIX (238): 433–460. doi:10.1093/mind/LIX.236.433.  https://dx.doi.org/10.1093%2Fmind%2FLIX.236.433
  23. Barricelli, Nils Aall (1954). "Esempi numerici di processi di evoluzione". Methodos: 45–68. 
  24. Barricelli, Nils Aall (1957). "Symbiogenetic evolution processes realized by artificial methods". Methodos: 143–182. 
  25. Fraser, Alex (1957). "Simulation of genetic systems by automatic digital computers. I. Introduction". Aust. J. Biol. Sci. 10: 484–491. doi:10.1071/BI9570484.  https://dx.doi.org/10.1071%2FBI9570484
  26. Del Moral, Pierre (2004). Feynman-Kac formulae. Genealogical and interacting particle approximations. Probability and its Applications. Springer. pp. 575. ISBN 9780387202686. https://www.springer.com/mathematics/probability/book/978-0-387-20268-6. "Series: Probability and Applications" 
  27. Del Moral, Pierre; Miclo, Laurent (2000). "Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering". Séminaire de Probabilités, XXXIV. Lecture Notes in Mathematics. 1729. pp. 1–145. doi:10.1007/bfb0103798. ISBN 978-3-540-67314-9. http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf. 
  28. Del Moral, Pierre; Miclo, Laurent (2000). "A Moran particle system approximation of Feynman-Kac formulae.". Stochastic Processes and Their Applications 86 (2): 193–216. doi:10.1016/S0304-4149(99)00094-0.  https://dx.doi.org/10.1016%2FS0304-4149%2899%2900094-0
  29. Del Moral, Pierre (2003). "Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups". ESAIM Probability & Statistics 7: 171–208. doi:10.1051/ps:2003001. http://journals.cambridge.org/download.php?file=%2FPSS%2FPSS7%2FS1292810003000016a.pdf&code=a0dbaa7ffca871126dc05fe2f918880a. 
  30. Assaraf, Roland; Caffarel, Michel; Khelif, Anatole (2000). "Diffusion Monte Carlo Methods with a fixed number of walkers". Phys. Rev. E 61 (4): 4566–4575. doi:10.1103/physreve.61.4566. PMID 11088257. Bibcode: 2000PhRvE..61.4566A. Archived from the original. Error: If you specify |archiveurl=, you must also specify |archivedate=. https://web.archive.org/web/20141107015724/http://qmcchem.ups-tlse.fr/files/caffarel/31.pdf. 
  31. Caffarel, Michel; Ceperley, David; Kalos, Malvin (1993). "Comment on Feynman-Kac Path-Integral Calculation of the Ground-State Energies of Atoms". Phys. Rev. Lett. 71 (13): 2159. doi:10.1103/physrevlett.71.2159. PMID 10054598. Bibcode: 1993PhRvL..71.2159C.  https://dx.doi.org/10.1103%2Fphysrevlett.71.2159
  32. Hetherington, Jack, H. (1984). "Observations on the statistical iteration of matrices". Phys. Rev. A 30 (2713): 2713–2719. doi:10.1103/PhysRevA.30.2713. Bibcode: 1984PhRvA..30.2713H.  https://dx.doi.org/10.1103%2FPhysRevA.30.2713
  33. Fermi, Enrique; Richtmyer, Robert, D. (1948). "Note on census-taking in Monte Carlo calculations". LAM 805 (A). http://scienze-como.uninsubria.it/bressanini/montecarlo-history/fermi-1948.pdf. "Declassified report Los Alamos Archive". 
  34. Rosenbluth, Marshall, N.; Rosenbluth, Arianna, W. (1955). "Monte-Carlo calculations of the average extension of macromolecular chains". J. Chem. Phys. 23 (2): 356–359. doi:10.1063/1.1741967. Bibcode: 1955JChPh..23..356R. https://semanticscholar.org/paper/1570c85ba9aca1cb413ada31e215e0917c3ccba7. 
  35. Gordon, N. J.; Salmond, D. J.; Smith, A. F. M. (1993). "Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proceedings F - Radar and Signal Processing 140 (2): 107–113. doi:10.1049/ip-f-2.1993.0015. https://ieeexplore.ieee.org/document/210672. Retrieved 2009-09-19. 
  36. Kitagawa, G. (1996). "Monte carlo filter and smoother for non-Gaussian nonlinear state space models". Journal of Computational and Graphical Statistics 5 (1): 1–25. doi:10.2307/1390750.  https://dx.doi.org/10.2307%2F1390750
  37. Carvalho, Himilcon; Del Moral, Pierre; Monin, André; Salut, Gérard (July 1997). "Optimal Non-linear Filtering in GPS/INS Integration.". IEEE Transactions on Aerospace and Electronic Systems 33 (3): 835. doi:10.1109/7.599254. Bibcode: 1997ITAES..33..835C. http://homepages.laas.fr/monin/Version_anglaise/Publications_files/GPS.pdf. 
  38. Del Moral, Pierre (1996). "Non Linear Filtering: Interacting Particle Solution.". Markov Processes and Related Fields 2 (4): 555–580. http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf. 
  39. 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).
  40. 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).
  41. 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).
  42. 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).
  43. 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).
  44. 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).
  45. Del Moral, Pierre (1996). "Non Linear Filtering: Interacting Particle Solution.". Markov Processes and Related Fields 2 (4): 555–580. http://web.maths.unsw.edu.au/~peterdel-moral/mprfs.pdf. 
  46. Del Moral, Pierre (1998). "Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems". Annals of Applied Probability 8 (2): 438–495. doi:10.1214/aoap/1028903535. http://projecteuclid.org/download/pdf_1/euclid.aoap/1028903535. 
  47. Crisan, Dan; Gaines, Jessica; Lyons, Terry (1998). "Convergence of a branching particle method to the solution of the Zakai". SIAM Journal on Applied Mathematics 58 (5): 1568–1590. doi:10.1137/s0036139996307371. https://semanticscholar.org/paper/99e8759a243cd0568b0f32cbace2ad0525b16bb6. 
  48. Crisan, Dan; Lyons, Terry (1997). "Nonlinear filtering and measure-valued processes". Probability Theory and Related Fields 109 (2): 217–244. doi:10.1007/s004400050131.  https://dx.doi.org/10.1007%2Fs004400050131
  49. Crisan, Dan; Lyons, Terry (1999). "A particle approximation of the solution of the Kushner–Stratonovitch equation". Probability Theory and Related Fields 115 (4): 549–578. doi:10.1007/s004400050249.  https://dx.doi.org/10.1007%2Fs004400050249
  50. Crisan, Dan; Del Moral, Pierre; Lyons, Terry (1999). "Discrete filtering using branching and interacting particle systems". Markov Processes and Related Fields 5 (3): 293–318. http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf. 
  51. Del Moral, Pierre; Guionnet, Alice (2001). "On the stability of interacting processes with applications to filtering and genetic algorithms". Annales de l'Institut Henri Poincaré 37 (2): 155–194. doi:10.1016/s0246-0203(00)01064-5. Bibcode: 2001AnIHP..37..155D. http://web.maths.unsw.edu.au/~peterdel-moral/ihp.ps. 
  52. Del Moral, Pierre; Guionnet, Alice (1999). "On the stability of Measure Valued Processes with Applications to filtering". C. R. Acad. Sci. Paris 39 (1): 429–434. 
  53. Malrieu, Florent (2001). "Logarithmic Sobolev inequalities for some nonlinear PDE's". Stochastic Process. Appl 95 (1): 109–132. doi:10.1016/s0304-4149(01)00095-3.  https://dx.doi.org/10.1016%2Fs0304-4149%2801%2900095-3
  54. Del Moral, Pierre (2013). Mean field simulation for Monte Carlo integration. Monographs on Statistics & Applied Probability. 126. ISBN 9781466504059. http://www.crcpress.com/product/isbn/9781466504059. 
  55. Del Moral, Pierre; Miclo, Laurent (2001). "Genealogies and Increasing Propagations of Chaos for Feynman-Kac and Genetic Models". Annals of Applied Probability 11 (4): 1166–1198. http://web.maths.unsw.edu.au/~peterdel-moral/spc.ps. 
  56. Del Moral, Pierre; Doucet, Arnaud; Singh, Sumeetpal, S. (2010). "A Backward Particle Interpretation of Feynman-Kac Formulae". M2AN 44 (5): 947–976. doi:10.1051/m2an/2010048. http://hal.inria.fr/docs/00/42/13/56/PDF/RR-7019.pdf. 
  57. Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2012). "On Adaptive Resampling Procedures for Sequential Monte Carlo Methods". Bernoulli 18 (1): 252–278. doi:10.3150/10-bej335. http://hal.inria.fr/docs/00/33/25/83/PDF/RR-6700.pdf. 
  58. Vergé, Christelle; Dubarry, Cyrille; Del Moral, Pierre; Moulines, Eric (2013). "On parallel implementation of Sequential Monte Carlo methods: the island particle model". Statistics and Computing 25 (2): 243–260. doi:10.1007/s11222-013-9429-x. Bibcode: 2013arXiv1306.3911V.  https://dx.doi.org/10.1007%2Fs11222-013-9429-x
  59. Chopin, Nicolas; Jacob, Pierre, E.; Papaspiliopoulos, Omiros (2011). "SMC^2: an efficient algorithm for sequential analysis of state-space models". arXiv:1101.1528v3 [stat.CO]. //arxiv.org/archive/stat.CO
  60. Andrieu, Christophe; Doucet, Arnaud; Holenstein, Roman (2010). "Particle Markov chain Monte Carlo methods". Journal of the Royal Statistical Society, Series B 72 (3): 269–342. doi:10.1111/j.1467-9868.2009.00736.x.  https://dx.doi.org/10.1111%2Fj.1467-9868.2009.00736.x
  61. Del Moral, Pierre; Patras, Frédéric; Kohn, Robert (2014). "On Feynman-Kac and particle Markov chain Monte Carlo models". arXiv:1404.5733 [math.PR]. //arxiv.org/archive/math.PR
  62. Cercignani, Carlo; Illner, Reinhard; Pulvirenti, Mario (1994). "The Mathematical Theory of Dilute Gases.". Springer. 
  63. Schrodinger, Erwin (1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules". Physical Review 28 (6): 1049–1070. doi:10.1103/physrev.28.1049. Bibcode: 1926PhRv...28.1049S.  https://dx.doi.org/10.1103%2Fphysrev.28.1049
  64. Del Moral, Pierre; Doucet, Arnaud (2004). "Particle Motions in Absorbing Medium with Hard and Soft Obstacles". Stochastic Analysis and Applications 22 (5): 1175–1207. doi:10.1081/SAP-200026444. http://web.maths.unsw.edu.au/~peterdel-moral/obstacle.ps. 
  65. Del Moral, Pierre; Doucet, Arnaud; Jasra, Ajay (2006). "Sequential Monte Carlo samplers". J. Royal Statist. Soc. B 68 (3): 411–436. doi:10.1111/j.1467-9868.2006.00553.x. http://web.maths.unsw.edu.au/~peterdel-moral/smc_samplers_try.pdf. 
  66. Lelièvre, Tony; Rousset, Mathias; Stoltz, Gabriel (2007). "Computation of free energy differences through nonequilibrium stochastic dynamics: the reaction coordinate case.". J. Comput. Phys. 222 (2): 624–643. doi:10.1016/j.jcp.2006.08.003. Bibcode: 2007JCoPh.222..624L.  https://dx.doi.org/10.1016%2Fj.jcp.2006.08.003
  67. Lelièvre, Tony; Rousset, Mathias; Stoltz, Gabriel (2010). "Free energy computations: A mathematical perspective". Imperial College Press: 472. 
  68. Caron, F.; Del Moral, P.; Pace, M.; Vo, B.-N. (2011). "On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering". Stochastic Analysis and Applications 29 (6): 951–997. doi:10.1080/07362994.2011.598797. ISSN 0736-2994.  https://dx.doi.org/10.1080%2F07362994.2011.598797
  69. Dynkin, Eugène, B. (1994). An Introduction to Branching Measure-Valued Processes. CRM Monograph Series. pp. 134. ISBN 978-0-8218-0269-4. 
  70. Zoia, Andrea; Dumonteil, Eric; Mazzolo, Alain (2012). "Discrete Feynman-Kac formulas for branching random walks". EPL 98 (40012): 40012. doi:10.1209/0295-5075/98/40012. Bibcode: 2012EL.....9840012Z.  https://dx.doi.org/10.1209%2F0295-5075%2F98%2F40012
  71. Caron, François; Del Moral, Pierre; Doucet, Arnaud; Pace, Michele (2011). "Particle approximations of a class of branching distribution flows arising in multi-target tracking". SIAM J. Control Optim.: 1766–1792. doi:10.1137/100788987. http://hal.archives-ouvertes.fr/docs/00/46/41/30/PDF/RR-7233.pdf. 
  72. Pitman, Jim; Fitzsimmons, Patrick, J. (1999). "Kac's moment formula and the Feynman–Kac formula for additive functionals of a Markov process". Stochastic Processes and Their Applications 79 (1): 117–134. doi:10.1016/S0304-4149(98)00081-7. http://math.ucsd.edu/~pfitz/downloads/kac/kac.html. 
  73. Arendt, Wolfgang; Batty, Charles, J.K. (1993). "Absorption semigroups and Dirichlet boundary conditions". Math. Ann. 295: 427–448. doi:10.1007/bf01444895. https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/downloads/pubbib/short/1993-AreBat-AbsSmgDrcBndCnd.pdf. 
  74. Lant, Timothy; Thieme, Horst (2007). "Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality". Positivity 11 (2): 299–318. doi:10.1007/s11117-006-2044-8.  https://dx.doi.org/10.1007%2Fs11117-006-2044-8
  75. Takeda, Masayoshi (2008). "Some Topics connected with Gaugeability for Feynman-Kac Functionals". RIMS Kokyuroku Bessatsu B6: 221–236. http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B6/pdf/B6-18.pdf. 
  76. Yaglom, Isaak (1947). "Certain limit theorems of the theory of branching processes". Dokl. Akad. Nauk SSSR 56: 795–798. 
  77. Del Moral, Pierre; Miclo, Laurent (2002). "On the Stability of Non Linear Semigroup of Feynman-Kac Type". Annales de la Faculté des Sciences de Toulouse 11 (2): 135–175. doi:10.5802/afst.1021. 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. 
  78. Kallel, Leila; Naudts, Bart; Rogers, Alex (2001-05-08). Theoretical Aspects of Evolutionary Computing. Springer, Berlin, New York; Natural computing series.. pp. 497. ISBN 978-3540673965. 
  79. Del Moral, Pierre; Kallel, Leila; Rowe, John (2001). "Modeling genetic algorithms with interacting particle systems". Revista de Matematica: Teoria y Aplicaciones 8 (2): 19–77. doi:10.15517/rmta.v8i2.201.  https://dx.doi.org/10.15517%2Frmta.v8i2.201
  80. Del Moral, Pierre; Guionnet, Alice (2001). "On the stability of interacting processes with applications to filtering and genetic algorithms". Annales de l'Institut Henri Poincaré 37 (2): 155–194. doi:10.1016/S0246-0203(00)01064-5. Bibcode: 2001AnIHP..37..155D.  https://dx.doi.org/10.1016%2FS0246-0203%2800%2901064-5
  81. Aumann, Robert John (1964). "Markets with a continuum of traders". Econometrica 32 (1–2): 39–50. doi:10.2307/1913732.  https://dx.doi.org/10.2307%2F1913732
  82. Jovanovic, Boyan; Rosenthal, Robert W. (1988). "Anonymous sequential games". Journal of Mathematical Economics 17 (1): 77–87. doi:10.1016/0304-4068(88)90029-8.  https://dx.doi.org/10.1016%2F0304-4068%2888%2990029-8
  83. Huang, Minyi.Y; Malhame, Roland P.; Caines, Peter E. (2006). "Large Population Stochastic Dynamic Games: Closed-Loop McKean–Vlasov Systems and the Nash Certainty Equivalence Principle". Communications in Information and Systems 6 (3): 221–252. doi:10.4310/CIS.2006.v6.n3.a5.  https://dx.doi.org/10.4310%2FCIS.2006.v6.n3.a5
  84. Maynard Smith, John (1982). Evolution and the Theory of Games. Cambridge University Press, Cambridge. 
  85. Kolokoltsov, Vassili; Li, Jiajie; Yang, Wei (2011). "Mean field games and nonlinear Markov processes". arXiv:1112.3744v2 [math.PR]. //arxiv.org/archive/math.PR
  86. Lasry, Jean Michel; Lions, Pierre Louis (2007). "Mean field games". Japanese J. Math 2 (1): 229–260. doi:10.1007/s11537-007-0657-8. https://basepub.dauphine.fr/handle/123456789/2263. 
  87. Carmona, René; Fouque, Jean Pierre; Sun, Li-Hsien (2014). "Mean Field Games and Systemic Risk". Communications in Mathematical Sciences. Bibcode: 2013arXiv1308.2172C.  http://adsabs.harvard.edu/abs/2013arXiv1308.2172C
  88. Budhiraja, Amarjit; Del Moral, Pierre; Rubenthaler, Sylvain (2013). "Discrete time Markovian agents interacting through a potential". ESAIM Probability & Statistics 17: 614–634. doi:10.1051/ps/2012014.  https://dx.doi.org/10.1051%2Fps%2F2012014
  89. Aumann, Robert (1964). "Markets with a continuum of traders". Econometrica 32 (1–2): 39–50. doi:10.2307/1913732. http://www.u.arizona.edu/~mwalker/501BReadings/Aumann1964.pdf. 
  90. Del Moral, Pierre; Lézaud, Pascal (2006). Branching and interacting particle interpretation of rare event probabilities. (stochastic Hybrid Systems: Theory and Safety Critical Applications, eds. H. Blom and J. Lygeros. ed.). Springer, Berlin. pp. 277–323. http://web.maths.unsw.edu.au/~peterdel-moral/Del-Moral-Lezaud-rare-events.pdf. 
  91. Crisan, Dan; Del Moral, Pierre; Lyons, Terry (1998). "Discrete Filtering Using Branching and Interacting Particle Systems.". Markov Processes and Related Fields 5 (3): 293–318. http://web.maths.unsw.edu.au/~peterdel-moral/crisan98discrete.pdf. 
  92. Crisan, Dan; Del Moral, Pierre; Lyons, Terry (1998). "Interacting Particle Systems Approximations of the Kushner Stratonovitch Equation". Advances in Applied Probability 31 (3): 819–838. doi:10.1239/aap/1029955206. http://web.maths.unsw.edu.au/~peterdel-moral/ks-approx.pdf. 
  93. Pace, Michele; Del Moral, Pierre (2013). "Mean-Field PHD Filters Based on Generalized Feynman-Kac Flow". IEEE Journal of Selected Topics in Signal Processing 7 (3): 484–495. doi:10.1109/JSTSP.2013.2250909. Bibcode: 2013ISTSP...7..484P.  https://dx.doi.org/10.1109%2FJSTSP.2013.2250909
  94. Cappe, O.; Moulines, E.; Ryden, T. (2005). Inference in Hidden Markov Models. Springer. 
  95. Liu, J. (2001). Monte Carlo strategies in Scientific Computing. Springer. 
  96. Doucet, A. (2001). de Freitas, J. F. G.; Gordon, J.. eds. Sequential Monte Carlo Methods in Practice. Springer. 
  97. Botev, Z. I.; Kroese, D. P. (2008). "Efficient Monte Carlo simulation via the generalized splitting method". Methodology and Computing in Applied Probability 10 (4): 471–505. doi:10.1007/s11009-008-9073-7.  https://dx.doi.org/10.1007%2Fs11009-008-9073-7
  98. Botev, Z. I.; Kroese, D. P. (2012). "Efficient Monte Carlo simulation via the generalized splitting method". Statistics and Computing 22 (1): 1–16. doi:10.1007/s11222-010-9201-4.  https://dx.doi.org/10.1007%2Fs11222-010-9201-4
  99. Cérou, Frédéric; Del Moral, Pierre; Furon, Teddy; Guyader, Arnaud (2012). "Sequential Monte Carlo for Rare event estimation". Statistics and Computing 22 (3): 795–808. doi:10.1007/s11222-011-9231-6. https://hal.inria.fr/inria-00584352/file/cdfg.pdf. 
  100. Kolokoltsov, Vassili (2010). Nonlinear Markov processes. Cambridge Univ. Press. pp. 375. 
More
©Text is available under the terms and conditions of the Creative Commons-Attribution ShareAlike (CC BY-SA) license; additional terms may apply. By using this site, you agree to the Terms and Conditions and Privacy Policy.
Upload a video for this entry
Information
Subjects: Others
Contributor MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to https://encyclopedia.pub/register :
HandWiki
View Times: 1.2K
Entry Collection: HandWiki
Revision: 1 time (View History)
Update Date: 26 Oct 2022
Table of Contents
    Notice
    You are not a member of the advisory board for this topic. If you want to update advisory board member profile, please contact office@encyclopedia.pub.
    OK
    Confirm
    Only members of the Encyclopedia advisory board for this topic are allowed to note entries. Would you like to become an advisory board member of the Encyclopedia?
    Yes
    No
    ${ textCharacter }/${ maxCharacter }
    Submit
    Cancel
    There is no comment~
    ${ textCharacter }/${ maxCharacter }
    Submit
    Cancel
    ${ selectedItem.replyTextCharacter }/${ selectedItem.replyMaxCharacter }
    Submit
    Cancel
    Confirm
    Are you sure to Delete?
    Yes No
    Academic Video Service
    Feedback