Continuous-time multi-type Ehrenfest model and related Ornstein-Uhlenbeck diffusion on a star graph

TL;DR

This paper introduces a continuous-time Ehrenfest model on a star graph, analyzes its behavior, and derives a diffusive approximation leading to an Ornstein-Uhlenbeck process with a truncated Gaussian stationary distribution.

Contribution

It extends the Ehrenfest model to a star graph with a detailed analysis and provides a diffusive approximation to an Ornstein-Uhlenbeck process on a spider domain.

Findings

The model's transient and asymptotic behaviors are characterized.

A diffusive approximation to the model is derived and analyzed.

The stationary distribution of the approximating process is a truncated Gaussian.

Abstract

We deal with a continuous-time Ehrenfest model defined over an extended star graph, defined as a lattice formed by the integers of $d$ semiaxis joined at the origin. The dynamics on each ray are regulated by linear transition rates, whereas the switching among rays at the origin occurs according to a general stochastic matrix. We perform a detailed investigation of the transient and asymptotic behavior of this process. We also obtain a diffusive approximation of the considered model, which leads to an Ornstein-Uhlenbeck diffusion process over a domain formed by semiaxis joined at the origin, named spider. We show that the approximating process possesses a truncated Gaussian stationary density. Finally, the goodness of the approximation is discussed through comparison of stationary distributions, means and variances.