Differential equation approximations of stochastic network processes: An operator semigroup approach

András Bátkai; Istvan Z. Kiss; Eszter Sikolya; Péter L. Simon; András Bátkai; Istvan Z. Kiss; Eszter Sikolya; Péter L. Simon

doi:10.3934/nhm.2012.7.43

Networks and Heterogeneous Media

2012, Volume 7, Issue 1: 43-58. doi: 10.3934/nhm.2012.7.43

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Differential equation approximations of stochastic network processes: An operator semigroup approach

1.
Loránd Eötvös University, Institute of Mathematics, Pázmány Péter Sétány 1C, H-1117 Budapest
2.
School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF

Received: 01 July 2011 Revised: 01 January 2012
Primary: 47D06; Secondary: 60J28, 92A15.

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size ($N$). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.
- Dynamic network,
- birth-and-death process,
- one-parameter operator semigroup,
- mean field approximation
Citation: András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach[J]. Networks and Heterogeneous Media, 2012, 7(1): 43-58. doi: 10.3934/nhm.2012.7.43

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Abstract

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size ($N$). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as $N$ tends to infinity. Using only elementary semigroup theory we can prove the order $\mathcal{O}(1/N)$ convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.

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