Non-Markovian diffusion equations and processes: analysis and simulation

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.