**Abstract : ** The rigorous fluid approximation of linear kinetic equations was first obtained in the late 70s when the equilibrium distribution
decays faster than polynomials. In this case the limit is a diffusion equation. In the case of heavy tail equilibrium distribution (with infinite variance),
the first rigorous derivation was obtained in 2011 in a joint paper with Mellet and Mischler, in the case of scattering operators.
The limit shows then *anomalous* diffusion;

it is a governed by a fractional diffusion equation. Lebeau and Puel proved last year
the first similar result for Fokker-Planck operator, in dimension 1 and assuming that the equilibrium distribution has finite mass.
Fournier and Tardif gave an alternative probabilistic proof, more general (covering any dimension and infinite-mass equilibrium distribution)
but non-constructive.
We present a unified elementary approach, fully quantitative, that covers all previous cases as well as new ones.
This is a joint work with Emeric Bouin (Université Paris-Dauphine).