Non-Markovian diffusion equations and processes: analysis and simulations

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.This work has been carried out in the framework of a research project for Fractional Calculus Modelling (URL: www.fracalmo.org). it was pursued while Antonio Mura was visiting Boston University as a recipient of a fellowship of the Marco Polo project of the University of Bologna. The authors appreciate partial support by the NSF Grants DMS-050547 and DMS-0706786 at Boston University, by the Italian Ministry of University (M.I.U.R) through the Research Commission of the University of Bologna, and by the National Institute of Nuclear Physics (INFN) through the Bologna branch (Theoretical Group). Finally, the authors would like to thank the anonymous referees for their comments. (DMS-050547 - NSF; DMS-0706786 - NSF; Italian Ministry of University (M.I.U.R); National Institute of Nuclear Physics (INFN))Accepted manuscrip