Intertwining and propagation of mixtures for generalized KMP models and harmonic models

TL;DR

This paper investigates a broad class of stochastic mass transport models, showing that their non-equilibrium steady states are mixtures of product measures with evolving mixing measures, extending previous results to more general graphs and models.

Contribution

It generalizes the understanding of invariant measures and the evolution of mixing measures in generalized KMP and harmonic models on complex graphs.

Findings

Mixture of product measures remains invariant under the dynamics.

The mixing measure evolves as a Markov process called the hidden parameter model.

Stationary distribution of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution.

Abstract

We study a class of stochastic models of mass transport on discrete vertex set $V$. For these models, a one-parameter family of homogeneous product measures $\otimes_{i\in V} \nu_\theta$ is reversible. We prove that the set of mixtures of inhomogeneous product measures with equilibrium marginals, i.e., the set of measures of the form \[ \int\Big(\bigotimes_{i\in V} \nu_{\theta_i}\Big) \,\Xi(\prod_{i\in V}d\theta_i) \] is left invariant by the dynamics in the course of time, and the ``mixing measure'' $\Xi$ evolves according to a Markov process which we then call ``the hidden parameter model''. This generalizes results from [7] to a larger class of models and on more general graphs. The class of models includes discrete and continuous generalized KMP models, as well as discrete and continuous harmonic models. The results imply that in all these models, the non-equilibrium steady state of their reservoir driven version is a mixture of product measures where the mixing measure is in turn the stationary state of the corresponding ``hidden parameter model''. For the boundary-driven harmonic models on the chain $\{1,\ldots, N\}$ with nearest neighbor edges, we recover that the stationary measure of the hidden parameter model is the joint distribution of the ordered Dirichlet distribution (cf. [3]), with a purely probabilistic proof based on a spatial Markov property of the hidden parameter model.