  In this paper we introduce and analyze a class of diffusion type equations
related to certain non-Markovian stochastic processes. We start from the
forward drift equation which is made non-local in time by the introduction of a
suitable chosen memory kernel K(t). The resulting non-Markovian equation can be
interpreted in a natural way as the evolution equation of the marginal density
function of a random time process l(t). We then consider the subordinated
process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding
time evolution of the marginal density function of Y(t) is governed by a
non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We
develop several applications and derive the exact solutions. We consider
different stochastic models for the given equations providing path simulations.
