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Mach, Tibor; Sturm, Anja; Swart, Jan M.

Recursive tree processes and the mean-field limit of stochastic flows. (English) Zbl 1446.82054

Electron. J. Probab. 25, Paper No. 61, 63 p. (2020).
Summary: Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by D. J. Aldous and A. Bandyopadhyay [Ann. Appl. Probab. 15, No. 2, 1047–1110 (2005; Zbl 1105.60012)] in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.

Cited in 4 Documents

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60J25 Continuous-time Markov processes on general state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

mean-field limit; recursive tree process; recursive distributional equation; endogeny; interacting particle systems; cooperative branching

Citations:

Zbl 1105.60012
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\textit{T. Mach} et al., Electron. J. Probab. 25, Paper No. 61, 63 p. (2020; Zbl 1446.82054)
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References:

[1] [AB05] Aldous, D.J. and Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15(2) (2005), 1047-1110. · Zbl 1105.60012
[2] [ADF18] Andreis, L., Dai Pra, P. and Fischer, M.: McKean-Vlasov limit for interacting systems with simultaneous jumps. Stoch. Anal. Appl. 36(6) (2018), 1-36. · Zbl 1420.60042
[3] [Ald00] Aldous, D.J.: The percolation process on a tree where infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc. 128 (2000), 465-477. · Zbl 0961.60096
[4] [Als12] Alsmeyer, G.: Random recursive equations and their distributional fixed points. Unpublished manuscript (2012), available from https://www.uni-muenster.de/Stochastik/ lehre/WS1112/StochRekGleichungenII/book.pdf.
[5] [BCH18] Baake, E., Cordero, F. and Hummel, S.: Lines of descent in the deterministic mutation-selection model with pairwise interaction, arXiv:1812.00872v2.
[6] [BDF20] Broutin, N., Devroye, L. and Fraiman, N.: Recursive functions on conditional Galton-Watson trees, Random Struct. Alg. (2020), 1-13.
[7] [Bou58] Bourbaki, N.: Éléments de Mathématique. VIII. Part. 1: Les Structures Fondamentales de l’Analyse. Livre III: Topologie Générale. Chap. 9: Utilisation des Nombres Réels en Topologie Générale. 2iéme éd. Actualités Scientifiques et Industrielles 1045 Hermann & Cie, Paris, 1958. · Zbl 0085.37103
[8] [BW97] Baake, E. and Wiehe, T.: Bifurcations in haploid and diploid sequence space models. J. Math. Biol. 35 (1997) 321-343. · Zbl 0866.92012
[9] [Cho69] Choquet, G.: Lectures on Analysis. Volume I. Integration and Topological Vector Spaces. Benjamin, London, 1969. · Zbl 0181.39601
[10] [DFL86] De Masi, A., Ferrari, P.A. and Lebowitz, J.L.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys. 44 (1986), 589-644. · Zbl 0629.60107
[11] [DN94] Durrett, R. and Neuhauser, C.: Particle systems and reaction-diffusion equations. Ann. Probab. 22(1) (1994), 289-333. · Zbl 0799.60093
[12] [DN97] Durrett, R. and Neuhauser, C.: Coexistence results for some competition models. Ann. Appl. Probab. 7(1) (1997), 10-45. · Zbl 0873.92020
[13] [EK86] Ethier, S.N. and Kurtz, T.G.: Markov Processes; Characterization and Convergence. John Wiley & Sons, New York, 1986. · Zbl 0592.60049
[14] [FL17] Foxall, E. and Lanchier, N.: Survival and extinction results for a patch model with sexual reproduction. J. Math. Biol. 74(6) (2017), 1299-1349. · Zbl 1362.92061
[15] [FM01] Fill, J.A. and Machida, M.: Stochastic monotonicity and realizable monotonicity Ann. Probab. 29(2) (2001), 938-978. · Zbl 1015.60010
[16] [HLL18] Hiai, F., Lawson, J. and Lim, Y.: The stochastic order of probability measures on ordered metric spaces J. Math. Anal. Appl. 464(1) (2018), 707-724. · Zbl 1397.60052
[17] [JPS20] Johnson, T., Podder, M. and Skerman, F.: Random tree recursions: which fixed points correspond to tangible sets of trees? Random Struct. Alg. 56(3) (2020), 796-837. · Zbl 1445.05028
[18] [KKO77] Kamae, T., Krengel, U. and O’Brien, G.L.: Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 (1977), 899-912. · Zbl 0371.60013
[19] [Lig85] Liggett, T.M.: Interacting Particle Systems. Springer, New York, 1985. · Zbl 0559.60078
[20] [Lin92] Lindvall, T.: Lectures on the Coupling Method. John Wiley and Sons, New York, 1992. · Zbl 0850.60019
[21] [Mac17] Mach, T.: Dualities and genealogies in stochastic population models. PhD thesis, University of Göttingen, 2017. · Zbl 1397.92003
[22] [MS19] Martin, J.B. and Stasiński, R.: Minimax functions on Galton-Watson trees, Combinatorics, Probability and Computing (2019), 1-30.
[23] [MSS18] Mach, T., Sturm, A. and Swart, J.M.: A new characterization of endogeny. Math. Phys. Anal. Geom. 21(4) (2018), paper no. 30, 19 pages. · Zbl 1402.60131
[24] [McK66] McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6) (1966), 1907-1911. · Zbl 0149.13501
[25] [Neu94] Neuhauser, C.: A long range sexual reproduction process. Stochastic Processes Appl. 53 (1994), 193-220. · Zbl 0810.60097
[26] [Nob92] Noble, C.: Equilibrium behavior of the sexual reproduction process with rapid diffusion. Ann. Probab. 20(2) (1992), 724-745. · Zbl 0757.60104
[27] [NP99] Neuhauser, C. and Pacala, S.W.: An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. 9(4) (1999), 1226-1259. · Zbl 0948.92022
[28] [Par05] Parthasarathy, K.R.: Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. (2005). · Zbl 1188.60001
[29] [RST19] Ráth, B., Swart, J.M., and Terpai, T.: Frozen percolation on the binary tree is nonendogenous, arXiv:1910.09213.
[30] [SS15] Sturm, A. and Swart, J.M.: A particle system with cooperative branching and coalescence. Ann. Appl. Probab. 25(3) (2015), 1616-1649. · Zbl 1327.82056
[31] [SS18] Sturm, A. and Swart, J.M.: Pathwise duals of monotone and additive Markov processes. J. Theor. Probab. 31(2) (2018), 932-983. · Zbl 1397.82039
[32] [ST85] Shiga, T. and Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69 (1985), 439-459. · Zbl 0607.60095
[33] [Str65] Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36 (1965), 423-439. · Zbl 0135.18701
[34] [Swa17] Swart, J.M.: A Course in Interacting Particle Systems, arXiv:1703.10007.
[35] [War06] Warren, J. · Zbl 1115.60045
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