  Local diffusivity of a protein depends crucially on the conformation, and the
conformational fluctuations are often non-Markovian. Here, we investigate the
Langevin equation with non-Markovian fluctuating diffusivity, where the
fluctuating diffusivity is modeled by a generalized Langevin equation under a
double-well potential. We find that non-Markovian fluctuating diffusivity
affect the global diffusivity, i.e., the diffusion coefficient obtained by the
long-time trajectories when the memory kernel in the generalized Langevin
equation is a power-law form. On the other hand, the diffusion coefficient does
not change when the memory kernel is exponential. We show that these
non-Markovian effects are the consequences of an everlasting effect of the
initial condition on the stationary distribution in the generalized Langevin
equation under a double-well potential due to long-term memory.
