Semi-Markov models and motion in heterogeneous media

Abstract

In this paper we study continuous time random walks (CTRWs) such that the holding time in each state has a distribution depending on the state itself. For such processes, we provide integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel. Particular attention is devoted to the case where the holding times have a power-law decaying density, whose exponent depends on the state itself, which leads to variable order fractional equations. A suitable limit yields a variable order fractional heat equation, which models anomalous diffusions in heterogeneous media.

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Cited by in corpus (4)

• On time-fractional diffusion equations with space-dependent variable order
• On Distributions of Certain State Dependent Fractional Point Processes
• From semi-Markov random evolutions to scattering transport and superdiffusion
• Asymptotic estimates of solutions to time-fractional diffusion equations with space-dependent variable order