# Discrete time markovian agents interacting through a potential

Amarjit Budhiraja; Pierre Del Moral; Sylvain Rubenthaler

ESAIM: Probability and Statistics (2013)

- Volume: 17, page 614-634
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topBudhiraja, Amarjit, Moral, Pierre Del, and Rubenthaler, Sylvain. "Discrete time markovian agents interacting through a potential." ESAIM: Probability and Statistics 17 (2013): 614-634. <http://eudml.org/doc/274350>.

@article{Budhiraja2013,

abstract = {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.},

author = {Budhiraja, Amarjit, Moral, Pierre Del, Rubenthaler, Sylvain},

journal = {ESAIM: Probability and Statistics},

keywords = {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},

language = {eng},

pages = {614-634},

publisher = {EDP-Sciences},

title = {Discrete time markovian agents interacting through a potential},

url = {http://eudml.org/doc/274350},

volume = {17},

year = {2013},

}

TY - JOUR

AU - Budhiraja, Amarjit

AU - Moral, Pierre Del

AU - Rubenthaler, Sylvain

TI - Discrete time markovian agents interacting through a potential

JO - ESAIM: Probability and Statistics

PY - 2013

PB - EDP-Sciences

VL - 17

SP - 614

EP - 634

AB - 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.

LA - eng

KW - 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

UR - http://eudml.org/doc/274350

ER -

## References

top- [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. Zbl1187.82067MR2594897
- [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. Zbl1238.60056MR2837574
- [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. Zbl1232.93083MR2847331
- [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. Iberoamericana19 (2003) 971–1018. Zbl1073.35127MR2053570
- [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. Zbl1082.76105MR2209130
- [7] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields140 (2008) 19–40. Zbl1169.35031MR2357669
- [8] D. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys.31 (1983) 29–85. MR711469
- [9] R.L. Dobrushin, Central limit theorem for nonstationary Markov chains I. Theory Probab. Appl.1 (1956) 65–80. Zbl0093.15001MR97112
- [10] R.L. Dobrushin, Central limit theorem for nonstationary Markov chains II. Theory Probab. Appl.1 (1956) 329–383. Zbl0093.15001
- [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. Zbl1025.35005MR1930708
- [12] J. Gartner, On the McKean-Vlasov limit for interacting diffusions. Math. Nachr.137 (1988) 197–248. Zbl0678.60100MR968996
- [13] K. Giesecke, K. Spiliopoulos and R. Sowers, Default clustering in large portfolios: typical events. Ann. Appl. Probab.23 (2013) 348–385. Zbl1262.91141MR3059238
- [14] C. Graham and P. Robert, Interacting multi-class transmissions in large stochastic networks. Ann. Appl. Probab.19 (2009) 2334–2361. Zbl1179.60067MR2588247
- [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. Zbl1259.91075
- [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. Zbl1031.60085MR1970276
- [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). Zbl1140.91012MR1996882
- [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. Zbl0732.60114MR1108185

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.