  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.
