Electronic Journal of Probability

Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes

Amarjit Budhiraja, Paul Dupuis, Markus Fischer, and Kavita Ramanan

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The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state exchangeable weakly interacting N -particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using previous results and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 81, 30 pp.

Accepted: 4 August 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 93D30: Scalar and vector Lyapunov functions 34D20: Stability 60F10: Large deviations 60K25: Queueing theory [See also 68M20, 90B22]

Nonlinear Markov processes weakly interact- ing particle systems interacting Markov chains mean field limit stability metastability Lyapunov functions relative entropy large deviations

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Budhiraja, Amarjit; Dupuis, Paul; Fischer, Markus; Ramanan, Kavita. Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes. Electron. J. Probab. 20 (2015), paper no. 81, 30 pp. doi:10.1214/EJP.v20-4004. https://projecteuclid.org/euclid.ejp/1465067187

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  • Antunes, Nelson; Fricker, Christine; Robert, Philippe; Tibi, Danielle. Stochastic networks with multiple stable points. Ann. Probab. 36 (2008), no. 1, 255–278.
  • Bertini, L.; De Sole, A.; Gabrielli, D.; Jona-Lasinio, G.; Landim, C. Macroscopic fluctuation theory for stationary non-equilibrium states. J. Statist. Phys. 107 (2002), no. 3-4, 635–675.
  • Bodineau, Thierry; Lebowitz, Joel; Mouhot, Clément; Villani, Cédric. Lyapunov functionals for boundary-driven nonlinear drift-diffusion equations. Nonlinearity 27 (2014), no. 9, 2111–2132.
  • V.S. Borkar and R. Sundaresan. Asymptotics of the invariant measure in mean field models with jumps. Stochastic Systems, 2:1–59, 2012.
  • A. Budhiraja, P. Dupuis, M. Fischer and K. Ramanan. Limits of relative entropies associated with weakly interacting particle systems. Electron. J. Probab. 20 (2015), no. 80, 1-22. DOI: 10.1214/EJP.v20-4003
  • Carrillo, José A.; McCann, Robert J.; Villani, Cédric. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003), no. 3, 971–1018.
  • Dupuis, Paul; Ellis, Richard S. A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1997. xviii+479 pp. ISBN: 0-471-07672-4.
  • P. Dupuis, K. Ramanan, and W. Wu. Sample path large deviation principle for mean field weakly interacting jump processes. In preparation., 2012.)0: 93–100, 2001.
  • Gibbens, R. J.; Hunt, P. J.; Kelly, F. P. Bistability in communication networks. Disorder in physical systems, 113–127, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990.
  • den Hollander, Frank. Large deviations. Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. x+143 pp. ISBN: 0-8218-1989-5.
  • Léonard, Christian. Large deviations for long range interacting particle systems with jumps. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 2, 289–323.
  • Spitzer, F. Random fields and interacting particle systems. Notes on lectures given at the 1971 MAA Summer Seminar, Williams College, Williamstown, Mass. Mathematical Association of America, Washington, D.C., 1971. i+126 pp. (not consecutively paged).
  • Tamura, Yozo. Free energy and the convergence of distributions of diffusion processes of McKean type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 443–484.
  • Veretennikov, A. Yu. On ergodic measures for McKean-Vlasov stochastic equations. Monte Carlo and quasi-Monte Carlo methods 2004, 471–486, Springer, Berlin, 2006.