# Semi-Markov models and motion in heterogeneous media

## TL;DR

This paper develops mathematical models for continuous time random walks in heterogeneous media, leading to variable order fractional equations that describe anomalous diffusion with state-dependent properties.

## Contribution

It introduces integro-differential equations with position-dependent kernels for CTRWs and derives variable order fractional equations for complex media.

## Key findings

- Derived Volterra-type integro-differential equations for CTRWs
- Established connection to variable order fractional heat equations
- Modeled anomalous diffusion in heterogeneous environments

## 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.