  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.
