REAL

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

Bátkai, András and Kiss, István Z. and Sikolya, Eszter and Simon L., Péter (2012) Differential equation approximations of stochastic network processes: An operator semigroup approach. NETWORKS AND HETEROGENEOUS MEDIA, 7 (1). pp. 43-58. ISSN 1556-1801

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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 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. © American Institute of Mathematical Sciences.

Item Type: Article
Additional Information: \n Loránd Eötvös University, Institute of Mathematics, Pázmány Péter Sétány 1C, H-1117 Budapest, Hungary \n School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, United Kingdom \n Cited By :8 \n Export Date: 13 November 2018 \n Correspondence Address: Bátkai, A.; Loránd Eötvös University, Institute of Mathematics, Pázmány Péter Sétány 1C, H-1117 Budapest, Hungary; email: batka@cs.elte.hu
Uncontrolled Keywords: Stochastic models; Markov processes; Ordinary differential equations; mean field approximation; mean field approximation; Dynamic network; Dynamic network; Semi-group; Stochastic networks; Birth-and-death process; One-parameter operator semigroup; Birth and death process; Countable systems; Mean-field equations; Semigroup approaches;
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 01 Oct 2020 10:06
Last Modified: 01 Oct 2020 10:06
URI: http://real.mtak.hu/id/eprint/115547

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