The Annals of Applied Probability

Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

Eric Luçon and Wilhelm Stannat

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Abstract

We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $\theta_{i}$ is attached to one site $x_{i}$ of a periodic lattice and the interaction between particles $\theta_{i}$ and $\theta_{j}$ decreases as $\vert x_{i}-x_{j}\vert^{-\alpha}$ for $\alpha\in[0,1)$. In a previous work [Ann. Appl. Probab. 24 (2014) 1946–1993], it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean–Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $\alpha\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-\alpha}$ in the case $\alpha\in(\frac{1}{2},1)$.

Article information

Source
Ann. Appl. Probab. Volume 26, Number 6 (2016), 3840-3909.

Dates
Received: February 2015
Revised: December 2015
First available in Project Euclid: 15 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1481792601

Digital Object Identifier
doi:10.1214/16-AAP1194

Mathematical Reviews number (MathSciNet)
MR3582819

Subjects
Primary: 60F05: Central limit and other weak theorems 60G57: Random measures
Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 92B25: Biological rhythms and synchronization

Keywords
Weakly interacting diffusions spatially-extended particle systems weighted empirical measures fluctuations Kuramoto model neuronal models stochastic partial differential equations

Citation

Luçon, Eric; Stannat, Wilhelm. Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction. Ann. Appl. Probab. 26 (2016), no. 6, 3840--3909. doi:10.1214/16-AAP1194. https://projecteuclid.org/euclid.aoap/1481792601


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References

  • [1] Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. and Spigler, R. (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Rev. Modern Phys. 77 137–185.
  • [2] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces, 2nd ed. Pure and Applied Mathematics (Amsterdam) 140. Elsevier, Amsterdam.
  • [3] Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M. (1988). Discontinuity of the magnetization in one-dimensional $1/\vert x-y\vert^{2}$ Ising and Potts models. J. Stat. Phys. 50 1–40.
  • [4] Aronson, D. G. (1968). Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa (3) 22 607–694.
  • [5] Arous, G. B. and Guionnet, A. (1995). Large deviations for Langevin spin glass dynamics. Probab. Theory Related Fields 102 455–509.
  • [6] Baladron, J., Fasoli, D., Faugeras, O. and Touboul, J. (2012). Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. J. Math. Neurosci. 2 Art. 10, 50.
  • [7] Ben Arous, G. and Guionnet, A. (1997). Symmetric Langevin spin glass dynamics. Ann. Probab. 25 1367–1422.
  • [8] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [9] Bolley, F., Guillin, A. and Malrieu, F. (2010). Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. ESAIM Math. Model. Numer. Anal. 44 867–884.
  • [10] Bossy, M. and Talay, D. (1996). Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation. Ann. Appl. Probab. 6 818–861.
  • [11] Bullmore, E. and Sporns, O. (2009). Complex brain networks: Graph theoretical analysis of structural and functional systems. Nat. Rev., Neurosci. 10 186–198.
  • [12] Chowdhury, D. and Cross, M. C. (2010). Synchronization of oscillators with long-range power law interactions. Phys. Rev. E 82 016205.
  • [13] Dai Pra, P. and den Hollander, F. (1996). McKean–Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84 735–772.
  • [14] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2015). Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25 2096–2133.
  • [15] Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics 35. Springer, New York.
  • [16] Fernandez, B. and Méléard, S. (1997). A Hilbertian approach for fluctuations on the McKean–Vlasov model. Stochastic Process. Appl. 71 33–53.
  • [17] Firpo, M.-C. and Ruffo, S. (2001). Chaos suppression in the large size limit for long-range systems. J. Phys. A 34 L511–L518.
  • [18] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • [19] Gärtner, J. (1988). On the McKean–Vlasov limit for interacting diffusions. Math. Nachr. 137 197–248.
  • [20] Giacomin, G., Luçon, E. and Poquet, C. (2014). Coherence stability and effect of random natural frequencies in populations of coupled oscillators. J. Dynam. Differential Equations 26 333–367.
  • [21] Godinho, D. and Quiñinao, C. (2015). Propagation of chaos for a subcritical Keller–Segel model. Ann. Inst. Henri Poincaré Probab. Stat. 51 965–992.
  • [22] Gupta, S., Potters, M. and Ruffo, S. (2012). One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes. Phys. Rev. E 85 066201.
  • [23] Joffe, A. and Métivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. in Appl. Probab. 18 20–65.
  • [24] Jourdain, B. and Méléard, S. (1998). Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. Henri Poincaré Probab. Stat. 34 727–766.
  • [25] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In École D’été de Probabilités de Saint-Flour, XII—1982. Lecture Notes in Math. 1097 143–303. Springer, Berlin.
  • [26] Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103–126.
  • [27] Kurtz, T. G. and Xiong, J. (2004). A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems. Commun. Math. Sci. 2 325–358.
  • [28] Luçon, E. (2011). Quenched limits and fluctuations of the empirical measure for plane rotators in random media. Electron. J. Probab. 16 792–829.
  • [29] Luçon, E. and Stannat, W. (2014). Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab. 24 1946–1993.
  • [30] Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 540–560.
  • [31] Maródi, M., d’Ovidio, F. and Vicsek, T. (2002). Synchronization of oscillators with long range interaction: Phase transition and anomalous finite size effects. Phys. Rev. E 66 011109.
  • [32] 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.
  • [33] Mitoma, I. (1985). An $\infty$-dimensional inhomogeneous Langevin’s equation. J. Funct. Anal. 61 342–359.
  • [34] Oelschläger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 458–479.
  • [35] Oelschläger, K. (1987). A fluctuation theorem for moderately interacting diffusion processes. Probab. Theory Related Fields 74 591–616.
  • [36] Omelchenko, I., Riemenschneider, B., Hövel, P., Maistrenko, Y. and Schöll, E. (2012). Transition from spatial coherence to incoherence in coupled chaotic systems. Phys. Rev. E 85 026212.
  • [37] Rogers, J. L. and Wille, L. T. (1996). Phase transitions in nonlinear oscillator chains. Phys. Rev. E 54 R2193–R2196.
  • [38] Strogatz, S. H. and Mirollo, R. E. (1991). Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63 613–635.
  • [39] Sznitman, A.-S. (1985). A fluctuation result for nonlinear diffusions. In Infinite-Dimensional Analysis and Stochastic Processes (Bielefeld, 1983) (S. Albeverio, ed.). Res. Notes in Math. 124 145–160. Pitman, Boston, MA.
  • [40] 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.
  • [41] Touboul, J. (2012). Limits and dynamics of stochastic neuronal networks with random heterogeneous delays. J. Stat. Phys. 149 569–597.
  • [42] Wainrib, G. and Touboul, J. (2013). Topological and dynamical complexity of random neural networks. Phys. Rev. Lett. 110 118101.