A Diffusion Limit for Markov Chains with Log-linear Interaction on a Graph

A.A. Puhalskii, V. Shcherbakov

2025, v.31, Issue 2, 127-142

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.

DOI:10.61102/1024-2953-mprf.2025.31.2.003

Keywords: interacting Markov chains, diffusion approximation, reflected Ornstein-Uhlenbeck process, reversibility, Skorohod problem

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