  We investigate Brownian motion with diffusivity alternately fluctuating
between fast and slow states. We assume that sojourn-time distributions of
these two states are given by exponential or power-law distributions. We
develop a theory of alternating renewal processes to study a relaxation
function which is expressed with an integral of the diffusivity over time. This
relaxation function can be related to a position correlation function if the
particle is in a harmonic potential, and to the self-intermediate scattering
function if the potential force is absent. It is theoretically shown that, at
short times, the exponential relaxation or the stretched-exponential relaxation
are observed depending on the power law index of the sojourn-time
distributions. In contrast, at long times, a power law decay with an
exponential cutoff is observed. The dependencies on the initial ensembles
(i.e., equilibrium or non-equilibrium initial ensembles) are also elucidated.
These theoretical results are consistent with numerical simulations.
