Annals of Applied Probability

Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime

Martin Hutzenthaler and Daniel Pieper

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are “of the same size”. In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this “sparse regime” the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees.

Article information

Source
Ann. Appl. Probab., Volume 30, Number 5 (2020), 2311-2354.

Dates
Received: April 2018
Revised: May 2019
First available in Project Euclid: 15 September 2020

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1600157076

Digital Object Identifier
doi:10.1214/20-AAP1559

Mathematical Reviews number (MathSciNet)
MR4149530

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D25: Population dynamics (general)

Keywords
Propagation of chaos interacting diffusions mean-field approximation many-demes limit measure-valued processes tree of excursions excursion measure altruistic defense

Citation

Hutzenthaler, Martin; Pieper, Daniel. Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime. Ann. Appl. Probab. 30 (2020), no. 5, 2311--2354. doi:10.1214/20-AAP1559. https://projecteuclid.org/euclid.aoap/1600157076


Export citation

References

  • [1] Buckdahn, R., Djehiche, B., Li, J. and Peng, S. (2009). Mean-field backward stochastic differential equations: A limit approach. Ann. Probab. 37 1524–1565.
  • [2] Chetwynd-Diggle, J. A. and Etheridge, A. M. (2018). SuperBrownian motion and the spatial Lambda-Fleming–Viot process. Electron. J. Probab. 23 Art. ID 71.
  • [3] Chetwynd-Diggle, J. A. and Klimek, A. (2019). Rare mutations in the spatial Lambda-Fleming–Viot model in a fluctuating environment and SuperBrownian motion. Available at arXiv:1901.04374.
  • [4] Cox, J. T. and Perkins, E. A. (2005). Rescaled Lotka–Volterra models converge to super-Brownian motion. Ann. Probab. 33 904–947.
  • [5] Dawson, D. A. and Greven, A. (2014). Spatial Fleming–Viot Models with Selection and Mutation. Lecture Notes in Math. 2092. Springer, Cham.
  • [6] Durrett, R. and Perkins, E. A. (1999). Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Related Fields 114 309–399.
  • [7] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
  • [8] Gärtner, J. (1988). On the McKean–Vlasov limit for interacting diffusions. Math. Nachr. 137 197–248.
  • [9] Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 256. Springer, Berlin.
  • [10] Hutzenthaler, M. (2009). The Virgin Island model. Electron. J. Probab. 14 1117–1161.
  • [11] Hutzenthaler, M. (2012). Interacting diffusions and trees of excursions: Convergence and comparison. Electron. J. Probab. 17 Art. ID 71.
  • [12] Hutzenthaler, M., Jordan, F. and Metzler, D. (2015). Altruistic defense traits in structured populations. Available at arXiv:1505.02154.
  • [13] Hutzenthaler, M. and Pieper, D. (2018). Differentiability of semigroups of stochastic differential equations with Hölder-continuous diffusion coefficients. Available at arXiv:1803.10608.
  • [14] Hutzenthaler, M. and Wakolbinger, A. (2007). Ergodic behavior of locally regulated branching populations. Ann. Appl. Probab. 17 474–501.
  • [15] Kac, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III 171–197. Univ. California Press, Berkeley, CA.
  • [16] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [17] Klenke, A. (2014). Probability Theory: A Comprehensive Course, 2nd ed. Universitext. Springer, London.
  • [18] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
  • [19] Liggett, T. M. (2010). Continuous Time Markov Processes: An Introduction. Graduate Studies in Mathematics 113. Amer. Math. Soc., Providence, RI.
  • [20] McKean, H. P. Jr. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967) 41–57. Air Force Office Sci. Res., Arlington, VA.
  • [21] Méléard, S. and Roelly-Coppoletta, S. (1987). A propagation of chaos result for a system of particles with moderate interaction. Stochastic Process. Appl. 26 317–332.
  • [22] Oelschläger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 458–479.
  • [23] Oelschläger, K. (1985). A law of large numbers for moderately interacting diffusion processes. Z. Wahrsch. Verw. Gebiete 69 279–322.
  • [24] Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425–457.
  • [25] Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 43–65.
  • [26] Shiga, T. and Shimizu, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 395–416.
  • [27] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [28] Wakeley, J. and Takahashi, T. (2004). The many-demes limit for selection and drift in a subdivided population. Theor. Popul. Biol. 66 83–91.
  • [29] Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155–167.