Electronic Journal of Probability

Recursive tree processes and the mean-field limit of stochastic flows

Tibor Mach, Anja Sturm, and Jan M. Swart

Full-text: Open access

Abstract

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 Aldous and Bandyopadyay 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.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 61, 63 pp.

Dates
Received: 30 December 2018
Accepted: 20 April 2020
First available in Project Euclid: 13 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1589335470

Digital Object Identifier
doi:10.1214/20-EJP460

Subjects
Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 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 [See also 82B43, 82C43]

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

Mach, Tibor; Sturm, Anja; Swart, Jan M. Recursive tree processes and the mean-field limit of stochastic flows. Electron. J. Probab. 25 (2020), paper no. 61, 63 pp. doi:10.1214/20-EJP460. https://projecteuclid.org/euclid.ejp/1589335470


Export citation

References

  • [AB05] Aldous, D.J. and Bandyopadhyay, A.: A survey of max-type recursive distributional equations. Ann. Appl. Probab. 15(2) (2005), 1047–1110.
  • [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.
  • [Ald00] Aldous, D.J.: The percolation process on a tree where infinite clusters are frozen. Math. Proc. Cambridge Philos. Soc. 128 (2000), 465–477.
  • [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.
  • [BCH18] Baake, E., Cordero, F. and Hummel, S.: Lines of descent in the deterministic mutation-selection model with pairwise interaction, arXiv:1812.00872v2.
  • [BDF20] Broutin, N., Devroye, L. and Fraiman, N.: Recursive functions on conditional Galton-Watson trees, Random Struct. Alg. (2020), 1–13.
  • [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.
  • [BW97] Baake, E. and Wiehe, T.: Bifurcations in haploid and diploid sequence space models. J. Math. Biol. 35 (1997) 321–343.
  • [Cho69] Choquet, G.: Lectures on Analysis. Volume I. Integration and Topological Vector Spaces. Benjamin, London, 1969.
  • [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.
  • [DN94] Durrett, R. and Neuhauser, C.: Particle systems and reaction-diffusion equations. Ann. Probab. 22(1) (1994), 289–333.
  • [DN97] Durrett, R. and Neuhauser, C.: Coexistence results for some competition models. Ann. Appl. Probab. 7(1) (1997), 10–45.
  • [EK86] Ethier, S.N. and Kurtz, T.G.: Markov Processes; Characterization and Convergence. John Wiley & Sons, New York, 1986.
  • [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.
  • [FM01] Fill, J.A. and Machida, M.: Stochastic monotonicity and realizable monotonicity Ann. Probab. 29(2) (2001), 938–978.
  • [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.
  • [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.
  • [KKO77] Kamae, T., Krengel, U. and O’Brien, G.L.: Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 (1977), 899–912.
  • [Lig85] Liggett, T.M.: Interacting Particle Systems. Springer, New York, 1985.
  • [Lin92] Lindvall, T.: Lectures on the Coupling Method. John Wiley and Sons, New York, 1992.
  • [Mac17] Mach, T.: Dualities and genealogies in stochastic population models. PhD thesis, University of Göttingen, 2017.
  • [MS19] Martin, J.B. and Stasiński, R.: Minimax functions on Galton-Watson trees, Combinatorics, Probability and Computing (2019), 1–30.
  • [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.
  • [McK66] McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6) (1966), 1907–1911.
  • [Neu94] Neuhauser, C.: A long range sexual reproduction process. Stochastic Processes Appl. 53 (1994), 193–220.
  • [Nob92] Noble, C.: Equilibrium behavior of the sexual reproduction process with rapid diffusion. Ann. Probab. 20(2) (1992), 724–745.
  • [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.
  • [Par05] Parthasarathy, K.R.: Probability measures on metric spaces. AMS Chelsea Publishing, Providence, RI. (2005).
  • [RST19] Ráth, B., Swart, J.M., and Terpai, T.: Frozen percolation on the binary tree is nonendogenous, arXiv:1910.09213.
  • [SS15] Sturm, A. and Swart, J.M.: A particle system with cooperative branching and coalescence. Ann. Appl. Probab. 25(3) (2015), 1616–1649.
  • [SS18] Sturm, A. and Swart, J.M.: Pathwise duals of monotone and additive Markov processes. J. Theor. Probab. 31(2) (2018), 932–983.
  • [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.
  • [Str65] Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36 (1965), 423–439.
  • [Swa17] Swart, J.M.: A Course in Interacting Particle Systems, arXiv:1703.10007.
  • [War06] Warren, J.: Dynamics and endogeny for recursive processes on trees. Stochastics 78(5) (2006), 327–341.