  We explore the diffusion process in the non-Markovian spatio-temporal
noise.%the escape rate problem in the non-Markovian spatio-temporal random
noise. There is a non-trivial short memory regime, i.e., the Markovian limit
characterized by a scaling relation between the spatial and temporal
correlation lengths. In this regime, a Fokker-Planck equation is derived by
expanding the trajectory around the systematic motion and the non-Markovian
nature amounts to the systematic reduction of the potential. For a system with
the potential barrier, this fact leads to the renormalization of both the
barrier height and collisional prefactor in the Kramers escape rate, with the
resultant rate showing a maximum at some scaling limit.
