Steady state representations for the harmonic process

TL;DR

This paper clarifies the connections between existing steady state solutions for the harmonic process and introduces a new matrix product representation not previously available.

Contribution

It derives a novel matrix product solution for the harmonic process's steady state, linking and extending known analytical forms.

Findings

Established the relation between different steady state representations

Provided the first matrix product solution for this model

Enhanced understanding of the harmonic process's steady state structure

Abstract

In this note we discuss how the matrix product solution for the steady state of the harmonic process is obtained from the solutions already known in the literature, i.e. the closed-form expression derived in arXiv:2107.01720 and the nested integral form obtained in arXiv:2307.02793 and arXiv:2307.14975. Our results clarify the relation between the three representations of the steady state and provide the matrix product solution that has not been available for this model before.