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Large-scale behavior of a particle system with mean-field interaction: Traveling wave solutions

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  27 September 2022

Alexander Stolyar Open the ORCID record for Alexander Stolyar [Opens in a new window]
Alexander Stolyar*
Affiliation:
University of Illinois at Urbana-Champaign
*
*Postal address: ISE Department and Coordinated Science Lab, University of Illinois at Urbana-Champaign, Urbana, IL 61801. Email address: stolyar@illinois.edu
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Abstract

We use probabilistic methods to study properties of mean-field models, which arise as large-scale limits of certain particle systems with mean-field interaction. The underlying particle system is such that n particles move forward on the real line. Specifically, each particle ‘jumps forward’ at some time points, with the instantaneous rate of jumps given by a decreasing function of the particle’s location quantile within the overall distribution of particle locations. A mean-field model describes the evolution of the particles’ distribution when n is large. It is essentially a solution to an integro-differential equation within a certain class. Our main results concern the existence and uniqueness of—and attraction to—mean-field models which are traveling waves, under general conditions on the jump-rate function and the jump-size distribution.

Keywords

Particle systemmean-field model dynamicsasymptotic behaviordistributed system synchronization

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 1 , March 2023 , pp. 245 - 274
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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