# Relaxation Functions of Ornstein-Uhlenbeck Process with Fluctuating Diffusivity

## TL;DR

This paper investigates the relaxation dynamics of an Ornstein-Uhlenbeck process with fluctuating diffusivity, providing a general theoretical framework and explicit solutions for specific models, revealing distinct relaxation behaviors.

## Contribution

It introduces a novel analytical approach to describe relaxation in OU processes with fluctuating diffusivity, extending understanding beyond traditional models.

## Key findings

- Relaxation functions expressed via transfer matrix eigenvalues.
- Analytic solutions for two-state and OU-type diffusivity models.
- Distinct relaxation behaviors compared to conventional models.

## Abstract

We study a relaxation behavior of an Ornstein-Uhlenbeck (OU) process with a time-dependent and fluctuating diffusivity. In this process, the dynamics of a position vector is modeled by the Langevin equation with a linear restoring force and a fluctuating diffusivity (FD). This process can be interpreted as a simple model of the relaxational dynamics with internal degrees of freedom or in a heterogeneous environment. By utilizing the functional integral expression and the transfer matrix method, we show that the relaxation function can be expressed in terms of the eigenvalues and eigenfunctions of the transfer matrix, for general FD processes. We apply our general theory to two simple FD processes, where the FD is described by the Markovian two-state model or an OU type process. We show analytic expressions of the relaxation functions in these models, and their asymptotic forms. We also show that the relaxation behavior of the OU process with an FD is qualitatively different from those obtained from conventional models such as the generalized Langevin equation.