The Annals of Probability

Stochastic networks with multiple stable points

Nelson Antunes, Christine Fricker, Philippe Robert, and Danielle Tibi

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Abstract

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Networks with a unique equilibrium point are also presented.

Article information

Source
Ann. Probab., Volume 36, Number 1 (2008), 255-278.

Dates
First available in Project Euclid: 28 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1196268679

Digital Object Identifier
doi:10.1214/009117907000000105

Mathematical Reviews number (MathSciNet)
MR2370604

Zentralblatt MATH identifier
1130.60086

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Energy function fixed point equations stable equilibrium points metastability mean field limit

Citation

Antunes, Nelson; Fricker, Christine; Robert, Philippe; Tibi, Danielle. Stochastic networks with multiple stable points. Ann. Probab. 36 (2008), no. 1, 255--278. doi:10.1214/009117907000000105. https://projecteuclid.org/euclid.aop/1196268679


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