Long-time behaviour of the correlated random walk system

TL;DR

This paper analyzes the long-term decay behavior of solutions to the correlated random walk system, revealing exponential decay rates linked to the dominant eigenvalue and providing a detailed spectral analysis.

Contribution

It offers a comprehensive spectral analysis of the correlated random walk system, including eigenvalues, eigenfunctions, and decay rates, which was not previously fully characterized.

Findings

Solutions decay exponentially with a rate given by the dominant eigenvalue.

The spectrum of the associated operator is fully described.

Eigenfunctions and their geometric properties are thoroughly analyzed.

Abstract

In this work, we consider the so-called correlated random walk system (also known as correlated motion or persistent motion system), used in biological modelling, among other fields, such as chromatography. This is a linear system which can also be seen as a weakly damped wave equation with certain boundary conditions. We are interested in the long-time behaviour of its solutions. To be precise, we will prove that the decay of the solutions to this problem is of exponential form, where the optimal decay rate exponent is given by the dominant eigenvalue of the corresponding operator. This eigenvalue can be obtained as a particular solution of a system of transcendental equations. A complete description of the spectrum of the operator is provided, together with a comprehensive analysis of the corresponding eigenfunctions and their geometry.