Matrix product approach for the asymmetric random average process

TL;DR

This paper derives explicit beta density representations for the steady states of an asymmetric random average process and introduces a matrix product ansatz to solve for the stationary states with continuous variables.

Contribution

It provides a rigorous derivation of local interaction functions and an exact matrix algebra solution for the process's steady states, advancing analytical methods for continuous-state stochastic models.

Findings

Explicit beta density functions for local interactions

Exact matrix algebra solution for steady states

Completion of a previously incomplete proof

Abstract

We consider the asymmetric random average process which is a one-dimensional stochastic lattice model with nearest neighbour interaction but continuous and unbounded state variables. First, the explicit functional representations, so-called beta densities, of all local interactions leading to steady states of product measure form are rigorously derived. This also completes an outstanding proof given in a previous publication. Then, we present an alternative solution for the processes with factorized stationary states by using a matrix product ansatz. Due to continuous state variables we obtain a matrix algebra in form of a functional equation which can be solved exactly.