The Annals of Applied Probability

Mean field limit for disordered diffusions with singular interactions

Eric Luçon and Wilhelm Stannat

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Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf{R} ^{m}$ in the presence of a random environment and with spatial extension: each diffusion is attached to one site of the lattice $\mathbf{Z} ^{d}$, and the interaction between two diffusions is attenuated by a spatial weight that depends on their positions. For a general class of singular weights (including the case already considered in the physical literature when interactions obey to a power-law of parameter $0<\alpha<d$), we address the convergence as $N\to\infty$ of the empirical measure of the diffusions to the solution of a deterministic McKean–Vlasov equation and prove well-posedness of this equation, even in the degenerate case without noise. We provide also precise estimates of the speed of this convergence, in terms of an appropriate weighted Wasserstein distance, exhibiting in particular nontrivial fluctuations in the power-law case when $\frac{d}{2}\leq\alpha<d$. Our framework covers the case of polynomially bounded monotone dynamics that are especially encountered in the main models of neural oscillators.

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Ann. Appl. Probab., Volume 24, Number 5 (2014), 1946-1993.

First available in Project Euclid: 26 June 2014

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures
Secondary: 60F15: Strong theorems 35Q92: PDEs in connection with biology and other natural sciences 82C44: Dynamics of disordered systems (random Ising systems, etc.)

Disordered models weakly interacting diffusions Wasserstein distance spatially extended particle systems dissipative systems Kuramoto model FitzHugh–Nagumo model


Luçon, Eric; Stannat, Wilhelm. Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab. 24 (2014), no. 5, 1946--1993. doi:10.1214/13-AAP968.

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  • [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] 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.
  • [3] Bertini, L., Giacomin, G. and Poquet, C. (2012). Synchronization and random long time dynamics for mean-field plane rotators. Available at arXiv:1209.4537.
  • [4] Bolley, F., Guillin, A. and Malrieu, F. (2010). Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov–Fokker–Planck equation. M2AN Math. Model. Numer. Anal. 44 867–884.
  • [5] Bolley, F., Guillin, A. and Villani, C. (2007). Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Related Fields 137 541–593.
  • [6] Bossy, M., Jabir, J.-F. and Talay, D. (2011). On conditional McKean Lagrangian stochastic models. Probab. Theory Related Fields 151 319–351.
  • [7] 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.
  • [8] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [9] Chowdhury, D. and Cross, M. C. (2010). Synchronization of oscillators with long-range power law interactions. Phys. Rev. E (3) 82 016205.
  • [10] Dai Pra, P. and den Hollander, F. (1996). McKean–Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84 735–772.
  • [11] Da Prato, G. and Tubaro, L. (1998). Some remarks about backward Itô formula and applications. Stoch. Anal. Appl. 16 993–1003.
  • [12] Delarue, F., Inglis, J., Rubenthaler, S. and Tanré, E. (2012). Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Available at arXiv:1211.0299.
  • [13] Del Moral, P. and Miclo, L. (2000). A Moran particle system approximation of Feynman–Kac formulae. Stochastic Process. Appl. 86 193–216.
  • [14] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • [15] Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics 35. Springer, New York.
  • [16] Gärtner, J. (1988). On the McKean–Vlasov limit for interacting diffusions. Math. Nachr. 137 197–248.
  • [17] Gel’fand, I. M. and Vilenkin, N. Y. (1964). Generalized Functions. Vol. 4: Applications of Harmonic Analysis. Academic Press, New York.
  • [18] Giacomin, G., Luçon, E. and Poquet, C. (2011). Coherence stability and effect of random natural frequencies in population of coupled oscillators. Available at arXiv:1111.3581.
  • [19] Giacomin, G., Pakdaman, K. and Pellegrin, X. (2012). Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators. Nonlinearity 25 1247–1273.
  • [20] 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 (3) 85 066201.
  • [21] Hairer, M., Hutzenthaler, M. and Jentzen, A. (2012). Loss of regularity for Kolmogorov equations. Available at arXiv:1209.6035.
  • [22] 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.
  • [23] Krylov, N. V. (1995). Introduction to the Theory of Diffusion Processes. Translations of Mathematical Monographs 142. Amer. Math. Soc., Providence, RI.
  • [24] Luçon, E. (2011). Quenched limits and fluctuations of the empirical measure for plane rotators in random media. Electron. J. Probab. 16 792–829.
  • [25] Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 540–560.
  • [26] 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 (3) 66 011109.
  • [27] 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.
  • [28] 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.
  • [29] Oelschläger, K. (1984). A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 458–479.
  • [30] Oelschläger, K. (1985). A law of large numbers for moderately interacting diffusion processes. Z. Wahrsch. Verw. Gebiete 69 279–322.
  • [31] Omelchenko, I., Maistrenko, Y., Hövel, P. and Schöll, E. (2011). Loss of coherence in dynamical networks: Spatial chaos and chimera states. Phys. Rev. Lett. 106 234102.
  • [32] 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 (3) 85 026212.
  • [33] Rogers, J. L. and Wille, L. T. (1996). Phase transitions in nonlinear oscillator chains. Phys. Rev. E (3) 54 R2193–R2196.
  • [34] Strogatz, S. H. and Mirollo, R. E. (1991). Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63 613–635.
  • [35] Sznitman, A.-S. (1984). Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56 311–336.
  • [36] 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.
  • [37] Touboul, J. (2011). Propagation of chaos in neural fields. Available at arXiv:1108.2414.
  • [38] Touboul, J. (2012). Limits and dynamics of stochastic neuronal networks with random heterogeneous delays. J. Stat. Phys. 149 569–597.
  • [39] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin.
  • [40] Wainrib, G. and Touboul, J. (2013). Topological and dynamical complexity of random neural networks. Phys. Rev. Lett. 110 118101.