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Continuum and thermodynamic limits for a simple random-exchange model
journal contribution
posted on 2023-06-10, 03:08 authored by Bertram Duering, Nicos GeorgiouNicos Georgiou, Sara Merino-Aceituno, Enrico ScalasWe 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.
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Publication status
- Published
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- Published version
Journal
Stochastic Processes and their ApplicationsISSN
0304-4149Publisher
ElsevierExternal DOI
Volume
149Page range
248-277Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2022-04-13First Open Access (FOA) Date
2022-04-13First Compliant Deposit (FCD) Date
2022-04-13Usage metrics
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