Düring, Bertram, Georgiou, Nicos, Merino-Aceituno, Sara and Scalas, Enrico (2022) Continuum and thermodynamic limits for a simple random-exchange model. Stochastic Processes and their Applications, 149 . pp. 248-277. doi:10.1016/j.spa.2022.03.015 ISSN 0304-4149. [ 🗎 Public]. [ (✓) hoa:511 ]

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Abstract

We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann’s program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system while the third is extraneous.

Item Type:	Journal Article
Subjects:	H Social Sciences > HB Economic Theory Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Wealth -- Moral and ethical aspects, Income distribution, Mean field theory, Markov processes, Limit theorems (Probability theory), Exponential functions, Stochastic models, Kinetic theory of gases
Journal or Publication Title:	Stochastic Processes and their Applications
Publisher:	Elsevier Science BV
ISSN:	0304-4149
Official Date:	July 2022
Dates:	Date Event July 2022 Published 4 April 2022 Available 25 March 2022 Accepted 14 September 2020 Submitted
Volume:	149
Page Range:	pp. 248-277
DOI:	10.1016/j.spa.2022.03.015
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	28 March 2022
Date of first compliant Open Access:	27 May 2022
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID RPG-2015-69 Leverhulme Trust http://dx.doi.org/10.13039/501100000275 EP/P021409/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 VRG17-014 Vienna Science and Technology Fund http://dx.doi.org/10.13039/501100001821 S18099 [JSPS] Japan Society for the Promotion of Science http://dx.doi.org/10.13039/501100001691

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