Discrete time markovian agents interacting through a potential
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 614-634.

A discrete time stochastic model for a multiagent system given in terms of a large collection of interacting Markov chains is studied. The evolution of the interacting particles is described through a time inhomogeneous transition probability kernel that depends on the ‘gradient' of the potential field. The particles, in turn, dynamically modify the potential field through their cumulative input. Interacting Markov processes of the above form have been suggested as models for active biological transport in response to external stimulus such as a chemical gradient. One of the basic mathematical challenges is to develop a general theory of stability for such interacting Markovian systems and for the corresponding nonlinear Markov processes that arise in the large agent limit. Such a theory would be key to a mathematical understanding of the interactive structure formation that results from the complex feedback between the agents and the potential field. It will also be a crucial ingredient in developing simulation schemes that are faithful to the underlying model over long periods of time. The goal of this work is to study qualitative properties of the above stochastic system as the number of particles (N) and the time parameter (n) approach infinity. In this regard asymptotic properties of a deterministic nonlinear dynamical system, that arises in the propagation of chaos limit of the stochastic model, play a key role. We show that under suitable conditions this dynamical system has a unique fixed point. This result allows us to study stability properties of the underlying stochastic model. We show that as N → ∞, the stochastic system is well approximated by the dynamical system, uniformly over time. As a consequence, for an arbitrarily initialized system, as N → ∞ and n → ∞, the potential field and the empirical measure of the interacting particles are shown to converge to the unique fixed point of the dynamical system. In general, simulation of such interacting Markovian systems is a computationally daunting task. We propose a particle based approximation for the dynamic potential field which allows for a numerically tractable simulation scheme. It is shown that this simulation scheme well approximates the true physical system, uniformly over an infinite time horizon.

DOI : https://doi.org/10.1051/ps/2012014
Classification : 60J05,  60K35,  92C45,  70K20,  60K40
Mots clés : interacting Markov chains, agent based modeling, multi-agent systems, propagation of chaos, non-linear Markov processes, stochastic algorithms, stability, particle approximations, swarm simulations, chemotaxis, reinforced random walk
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     author = {Budhiraja, Amarjit and Moral, Pierre Del and Rubenthaler, Sylvain},
     title = {Discrete time markovian agents interacting through a potential},
     journal = {ESAIM: Probability and Statistics},
     pages = {614--634},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2012014},
     mrnumber = {3126154},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2012014/}
}
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Budhiraja, Amarjit; Moral, Pierre Del; Rubenthaler, Sylvain. Discrete time markovian agents interacting through a potential. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 614-634. doi : 10.1051/ps/2012014. http://www.numdam.org/articles/10.1051/ps/2012014/

[1] N. Bartoli and P. Del Moral, Simulation & Algorithmes Stochastiques. Cépaduès éditions (2001).

[2] L. Bertini, G. Giacomin and K. Pakdaman, Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Stat. Phys. 138 (2010) 270-290. | MR 2594897 | Zbl 1187.82067

[3] F. Caron, P. Del Moral, A. Doucet and M. Pace, Particle approximations of a class of branching distribution arising in multi-target tracking. SIAM J. Contr. Optim. 49 (2011) 1766-1792. | MR 2837574 | Zbl 1238.60056

[4] F. Caron, P. Del Moral, M. Pace and B.-N. Vo, On the stability and the approximation of branching distribution flows, with applications to nonlinear multiple target filtering. Stoch. Anal. Appl. 29 (2011) 951-997. | MR 2847331 | Zbl 1232.93083

[5] J. Carrillo, R. Mccann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003) 971-1018. | MR 2053570 | Zbl 1073.35127

[6] J. Carrillo, R. Mccann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179 (2006) 217-263. | MR 2209130 | Zbl 1082.76105

[7] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140 (2008) 19-40. | MR 2357669 | Zbl 1169.35031

[8] D. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31 (1983) 29-85. | MR 711469

[9] R.L. Dobrushin, Central limit theorem for nonstationary Markov chains I. Theory Probab. Appl. 1 (1956) 65-80. | MR 97112 | Zbl 0093.15001

[10] R.L. Dobrushin, Central limit theorem for nonstationary Markov chains II. Theory Probab. Appl. 1 (1956) 329-383. | Zbl 0093.15001

[11] A. Friedman and J.I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. Appl. 272 (2002) 138-163. | MR 1930708 | Zbl 1025.35005

[12] J. Gartner, On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 (1988) 197-248. | MR 968996 | Zbl 0678.60100

[13] K. Giesecke, K. Spiliopoulos and R. Sowers, Default clustering in large portfolios: typical events. Ann. Appl. Probab. 23 (2013) 348-385. | MR 3059238 | Zbl 1262.91141

[14] C. Graham and P. Robert, Interacting multi-class transmissions in large stochastic networks. Ann. Appl. Probab. 19 (2009) 2334-2361. | MR 2588247 | Zbl 1179.60067

[15] C.l Graham, J. Gomez-Serrano and J. Yves Le Boudec, The bounded confidence model of opinion dynamics (2010) arxiv.org/pdf/1006.3798. | Zbl 1259.91075

[16] B. Latané and A. Nowak, Self-organizing social systems: Necessary and sufficient conditions for the emergence of clustering, consolidation, and continuing diversity. Prog. Commun. Sci. (1997) 43-74.

[17] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 (2003) 540-560. | MR 1970276 | Zbl 1031.60085

[18] F. Schweitzer, Brownian agents and active particles. Springer Series in Synergetics. Collective dynamics in the natural and social sciences, With a foreword by J. Doyne Farmer. Springer-Verlag, Berlin (2003). | MR 1996882 | Zbl 1140.91012

[19] A. Stevens, Trail following and aggregation of myxobacteria. J. Biol. Syst. 3 (1995) 1059-1068.

[20] A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, vol. 1464 of Lect. Notes Math. Springer, Berlin (1991) 165-251. | MR 1108185 | Zbl 0732.60114

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