A diffusion limit for Markov chains with log-linear interaction on a graph

TL;DR

This paper proves that a multivariate Markov chain with log-linear interactions on a graph converges to a system of reflected Ornstein-Uhlenbeck processes under proper scaling, linking queueing theory and diffusion limits.

Contribution

It establishes a diffusion limit for Markov chains with graph-based log-linear interactions, extending heavy traffic queueing results to a new class of models.

Findings

Convergence of scaled Markov chains to reflected Ornstein-Uhlenbeck processes.

Diffusion limit applies when component values expand to all non-negative integers.

Methodology combines queueing theory and martingale techniques.

Abstract

In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods.