On relation between generalized diffusion equations and subordination schemes

TL;DR

This paper explores the relationship between generalized diffusion equations and subordination schemes, identifying conditions for their equivalence and applicability to various non-Fickian diffusion processes.

Contribution

It clarifies when generalized diffusion equations correspond to subordination schemes and provides examples of processes fitting only one or both descriptions.

Findings

Conditions for equivalence between diffusion equations and subordination schemes

Examples of processes with only one or both descriptions

Insights into modeling non-Fickian diffusion processes

Abstract

Generalized (non-Markovian) diffusion equations with different memory kernels and subordination schemes based on random time change in the Brownian diffusion process are popular mathematical tools for description of a variety of non-Fickian diffusion processes in physics, biology and earth sciences. Some of such processes (notably, the fluid limits of continuous time random walks) allow for either kind of description, but other ones do not. In the present work we discuss the conditions under which a generalized diffusion equation does correspond to a subordination scheme, and the conditions under which a subordination scheme does possess the corresponding generalized diffusion equation. Moreover, we discuss examples of random processes for which only one, or both kinds of description are applicable.