Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses

TL;DR

This paper connects the stationary distribution of an asymmetric exclusion process with open boundaries to quadratic harness processes, deriving explicit formulas and large deviation principles.

Contribution

It introduces a novel representation of the stationary measure using quadratic harnesses, enabling explicit calculations and asymptotic analysis.

Findings

Derived explicit formulas for site occupancy.

Proved large deviations principle for total particles.

Established connections between exclusion processes and quadratic harnesses.

Abstract

We show that the joint probability generating function of the stationary measure of a finite state asymmetric exclusion process with open boundaries can be expressed in terms of joint moments of Markov processes called quadratic harnesses. We use our representation to prove the large deviations principle for the total number of particles in the system. We use the generator of the Markov process to show how explicit formulas for the average occupancy of a site arise for special choices of parameters. We also give similar representations for limits of stationary measures as the number of sites tends to infinity.