Continuum and thermodynamic limits for a simple random-exchange model

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 and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.

Figures (2)

• Figure 1: Commutative diagram demonstrating the various limiting measures, depending on the order limits are taken, when the total wealth remains constant. Measures $\mu^{n, N}_{\infty}$ and $\mu^{\infty, N}_{\infty}$ denote the invariant distributions for the two Markov chains respectively.
• Figure 2: Commutative diagram demonstrating the various limiting measures, depending on the order limits are taken, when the total wealth remains constant. There are two parameters that scale; the number of agents $N$ and the time $t$. Time is discrete for the left down-arrow, but continuous in the right down-arrow. There is an intermediate step missing from the diagram in which discrete time events are changed with time events arising from a Poisson process of rate $1/N$ which simultaneously scales with $N$. That is called the Poissonisation step, and when the mean-field limits (M-F) are taken, the rate of the Poisson process also scales with $N$.

Theorems & Definitions (23)

• Remark 2.2
• Remark 2.3
• proof : Proof of Theorem \ref{['thm:processlevel']}
• Remark 3.1: Almost sure convergence for finite sample paths
• proof
• proof : Proof of Lemma \ref{['lem:tightness_for_the_action']}
• proof : Proof of Lemma \ref{['lem:tightness_for_the_measures']}
• proof : Proof of Proposition \ref{['prop:almost_sure_convergence_for_measures']}
• proof : Proof of Lemma \ref{['lem:continuity_of_limit']}
• proof : Proof of Lemma \ref{['lem:uniform_convergence_in_time']}