  The non-Debye, \textit{i.e.,} non-exponential, behavior characterizes a large
plethora of dielectric relaxation phenomena. Attempts to find their theoretical
explanation are dominated either by considerations rooted in the stochastic
processes methodology or by the so-called \textsl{fractional dynamics} based on
equations involving fractional derivatives which mimic the non-local time
evolution and as such may be interpreted as describing memory effects. Using
the recent results coming from the stochastic approach we link memory functions
with the Laplace (characteristic) exponents of infinitely divisible probability
distributions and show how to relate the latter with experimentally measurable
spectral functions characterizing relaxation in the frequency domain. This
enables us to incorporate phenomenological knowledge into the evolution laws.
To illustrate our approach we consider the standard Havriliak-Negami and
Jurlewicz-Weron-Stanislavsky models for which we derive well-defined evolution
equations. Merging stochastic and fractional dynamics approaches sheds also new
light on the analysis of relaxation phenomena which description needs going
beyond using the single evolution pattern. We determine sufficient conditions
under which such description is consistent with general requirements of our
approach.
