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Propagation of chaos and large deviations in mean-field models with jumps on block-structured networks

Part of: Limit theorems Markov processes Special processes Graph theory

Published online by Cambridge University Press:  05 June 2023

Donald A. Dawson ,
Ahmed Sid-Ali  and
Yiqiang Q. Zhao
Donald A. Dawson*
Affiliation:
Carleton University
Ahmed Sid-Ali*
Affiliation:
Carleton University
Yiqiang Q. Zhao*
Affiliation:
Carleton University
*
*Postal address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada.
*Postal address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada.
*Postal address: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada.
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Abstract

A system of interacting multi-class finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-color nodes. The interactions are local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to nodes from the same block, whereas the peripheral nodes are connected to both nodes from the same block and nodes from other blocks. Limits of such systems as the number of nodes tends to infinity are investigated. In particular, under specific regularity conditions, propagation of chaos and the law of large numbers are established in a multi-population setting. Moreover, it is shown that, as the number of nodes goes to infinity, the behavior of the system can be represented by the solution of a McKean–Vlasov system. Then, we prove large deviations principles for the vectors of empirical measures and the empirical processes, which extends the classical results of Dawson and Gärtner (Stochastics 20, 1987) and Léonard (Ann. Inst. H. Poincaré Prob. Statist. 31, 1995).

Keywords

Interacting particle systemsinhomogeneous graphsMcKean–Vlasov equationmulti-class populationsjump processes

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J27: Continuous-time Markov processes on discrete state spaces 60F10: Large deviations
Secondary: 05C82: Small world graphs, complex networks
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 4 , December 2023 , pp. 1301 - 1361
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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