Duality and stationary distributions of the "Immediate Exchange Model" and its generalizations

TL;DR

This paper explores the duality and stationary distributions of the Immediate Exchange Model and its generalizations, revealing new invariance and ergodicity properties through duality functions and symmetry analysis.

Contribution

It introduces a discrete dual for the model, establishes invariance of Gamma product measures, and extends results to a generalized Beta-distributed exchange fraction.

Findings

Invariance of Gamma distribution products with shape 2

Discrete duality with self-duality property

SU(1,1) symmetry in continuous and discrete models

Abstract

We prove that the "Immediate Exchange Model" of econophysics has a discrete dual, where the duality functions are those connecting the Brownian Energy Process and the Symmetric Inclusion Process. As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties of a generalized model, where the exchange fraction is $Beta(s,t)$ distributed (instead of uniform), and product measures with $\mbox{Gamma}(s+t)$ marginals are invariant. We prove that the discrete dual has the self-duality property, and prove full SU(1,1) for both the continuous and discrete model.