Stochastic Systems

Asymptotics of the invariant measure in mean field models with jumps

Vivek S. Borkar and Rajesh Sundaresan

Full-text: Open access

Abstract

We consider the asymptotics of the invariant measure for the process of the empirical spatial distribution of $N$ coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of the transition rates on this spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. It is also applicable to the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution asymptotically concentrates on this equilibrium. More generally, its limit points are supported on a subset of the $\omega$-limit sets of the McKean-Vlasov equation. Using a control-theoretic approach, we examine the question of large deviations of the invariant measure from this limit.

Article information

Source
Stoch. Syst., Volume 2, Number 2 (2012), 322-380.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.ssy/1393252026

Digital Object Identifier
doi:10.1214/12-SSY064

Mathematical Reviews number (MathSciNet)
MR3354770

Zentralblatt MATH identifier
1296.60258

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F10: Large deviations 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B18: Communication networks [See also 68M10, 94A05] 49J15: Optimal control problems involving ordinary differential equations 34H05: Control problems [See also 49J15, 49K15, 93C15]

Keywords
Decoupling approximation fluid limit invariant measure McKean-Vlasov equation mean field limit small noise limit stationary measure stochastic Liouville equation

Citation

Borkar, Vivek S.; Sundaresan, Rajesh. Asymptotics of the invariant measure in mean field models with jumps. Stoch. Syst. 2 (2012), no. 2, 322--380. doi:10.1214/12-SSY064. https://projecteuclid.org/euclid.ssy/1393252026


Export citation

References

  • [1] Anantharam, V. (1991) “A mean field limit for a lattice caricature of dynamic routing in circuit switched networks”, Annals of Appl. Prob. 1, 481–503.
  • [2] Anantharam, V. and Benchekroun, M. (1993) “A technique for computing sojourn times in large networks of interacting queues”, Probability in the Engineering and Informational Sciences 7, 441–464.
  • [3] Benaim, M. and Le Boudec J.-Y. (2008) “A class of mean field interaction models for computer and communication systems”, Perform. Eval. 65, 823–838.
  • [4] Benaim, M. and Weibull, J. (2003) “Deterministic approximation of stochastic evolution”, Econometrica 71, 873-904.
  • [5] Bianchi, G. (1998) “IEEE 802.11 - saturated throughput analysis”, IEEE Comm. Lett. 12, 318–320.
  • [6] Biswas, A. and Borkar, V. S. (2009) “Small noise asymptotics for invariant densities for a class of diffusions: a control theoretic view”, Journal of Mathematical Analysis and Applications 360, 476–484. Correction note: arXiv:1107.2277v1
  • [7] Borkar, V. S. (1995) Probability Theory: An Advanced Course, Springer Verlag, New York.
  • [8] Bordenave, C., McDonald, D. and Proutière, A. (2010) “A particle system in interaction with a rapidly varying environment: mean field limits and applications”, Networks and Heterogeneous Media, 5(1).
  • [9] Bordenave, C., McDonald, D. and Proutière, A. (2005) “Random multi-access algorithms, a mean field analysis”, Proc. 43th Allerton conference.
  • [10] Dai Pra, P. and den Hollander, F. (1996), “McKean-Vlasov limit for interacting random processes in random media”, Journal of Statistical Physics, 84(3/4), 735–772.
  • [11] Dawson, D. A. and Gärtner, J. (1987) “Large deviations from the McKean-Vlasov limit for weakly interacting diffusions”, Stochastics, 20, 247–308.
  • [12] Day, M. V. (1987) “Recent progress on the small parameter exit problem”, Stochastics, 20, 121–150.
  • [13] Dembo, A. and Zeitouni, O. (2010) Large Deviations Techniques and Applications, (2nd ed., corrected printing), Springer-Verlag, Berlin Heidelberg.
  • [14] Del Moral, P. and Zajic, T. (2003) “Note on the Laplace-Varadhan Integral Lemma”, Bernoulli, 9(1), Feb. 2003, 49–65.
  • [15] Djehiche, B. and Kaj, I. (1995) “The rate function for some measure-valued jump processes”, The Annals of Probability, 23(3), 1414–1438.
  • [16] Duffy, K. R. (2010) “Mean field Markov models of wireless local area networks”, Markov Processes and Related Fields 16(2), 295–328.
  • [17] Ethier, S. N. and Kurtz, T. G. (2005) Markov Processes: Characterization and Convergence (2nd ed.), John Wiley, New York.
  • [18] Feng, J. and Kurtz, T. G. (2006) Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs, 131. American Math. Soc., Providence, RI.
  • [19] Feng, S. (1994) “Large deviations for Markov processes with mean field interaction and unbounded jumps”, Probability Th. and Related Fields 100, 227–252.
  • [20] Feng, S. (1994) “Large deviations for empirical process of mean-field interacting particle system with unbounded jumps”, The Annals of Probability, 22(4), 2122–2151.
  • [21] Freidlin, M. I. and Wentzell, A. D. (1998) Random Perturbations of Dynamical Systems (2nd ed.), Springer Verlag, Berlin–Heidelberg.
  • [22] Graham, C. (2000) “Chaoticity on path space for a queueing network with selection of the shortest queue among several”, J. Appl. Probab. 37(1), 198–211.
  • [23] Kac, M. (1956) “Foundations of kinetic theory”, in ‘Proc. of 3rd Berkeley Symp. on Math. Stat. and Prob.’ (J. Neyman, ed.), Uni. of California Press, 171–197.
  • [24] Léonard, C. (1995) “Large deviations for long range interacting particle systems with jumps”, Annales de l’I. H. P., section B, 31(2), 289–323.
  • [25] McKean, H. P. (1966) “A class of Markov processes associated with nonlinear parabolic equations”, Proc. Nat. Acad. Sci. 56, 1907–1911.
  • [26] Ramaiyan, V., Kumar, A. and Altman, E. (2008) “Fixed point analysis of single cell IEEE 802.11e WLANs: uniqueness and multistability”, IEEE/ACM Trans. Network. 16, 1080–1093.
  • [27] Sharma, G., Ganesh, A. J. and Key, P. (2009) “Performance analysis of contention based medium access protocols”, IEEE Trans. Information Th. 55(4), 1665–1682. Also reported in Proc. IEEE Infocom (2006).
  • [28] Sheu, S-J. (1985) “Stochastic control and exit probabilities of jump processes”, SIAM J. Control and Optimization 23(2), 306–328, March 1985.
  • [29] Sheu, S-J. (1986) “Asymptotic behavior of the invariant density of a diffusion Markov process with small diffusion”, SIAM J. Math. Anal. 17(2), 451–460.
  • [30] Shwartz, A. and Weiss, A. (1995) Large Deviations for Performance Analysis, Chapman & Hall.
  • [31] Stolyar, A. L. (1989) “Asymptotic behavior of the stationary distribution for a closed queueing system”, Problemy Peredachi Informatsii, 25(4), 80–92.
  • [32] Sznitman, A. S. (1991) “Topics in propogation of chaos”, in ‘Ecole d’été de Probabilités de Saint Flour (1989)’, Springer Lecture Notes in Mathematics No. 1464, 166–251
  • [33] Vvedenskaya, N. D. and Sukhov Yu. M. (2007), “A multi-user multiple-access system: stability and metastability”, Problemy Peredachi Informatsii 43, 105–111.