Relation between generalized diffusion equations and subordination schemes

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.

Automated summary

Generalized diffusion equations are popular tools for describing non-Fickian diffusion processes, but not all processes can be described by both schemes. This work discusses the conditions for correspondence between a generalized diffusion equation and a subordination scheme, as well as examples of random processes that are applicable to one or both descriptions.