Non-Markovian diffusion equations and processes: Analysis and simulations
Taqqu, Murad S.
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Citation (published version)A. Mura, M.S. Taqqu, F. Mainardi. 2008. "Non-Markovian diffusion equations and processes: Analysis and simulations." PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS, Volume 387, Issue 21, pp. 5033 - 5064 (32). https://doi.org/10.1016/j.physa.2008.04.035
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.