  In this paper, we investigate the spectral analysis (from the point of view
of semi-groups) of discrete, fractional and classical Fokker-Planck equations.
Discrete and fractional Fokker-Planck equations converge in some sense to the
classical one. As a consequence, we first deal with discrete and classical
Fokker-Planck equations in a same framework, proving uniform spectral estimates
using a perturbation argument and an enlargement argument. Then, we do a
similar analysis for fractional and classical Fokker-Planck equations using an
argument of enlargement of the space in which the semigroup decays. We also
handle another class of discrete Fokker-Planck equations which converge to the
fractional Fokker-Planck one, we are also able to treat these equations in a
same framework from the spectral analysis viewpoint, still with a semigroup
approach and thanks to a perturbative argument combined with an enlargement
one. Let us emphasize here that we improve the perturbative argument introduced
in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging
thus the class of operators which fulfills the assumptions required to apply
it.
