  This paper is devoted to kinetic equations without confinement. We
investigate the large time behaviour induced by collision operators with fat
tailed local equilibria. Such operators have an anomalous diffusion limit. In
the appropriate scaling, the macroscopic equation involves a fractional
diffusion operator so that the optimal decay rate is determined by a fractional
Nash type inequality. At kinetic level we develop an $\mathrm
L^2$-hypocoercivity approach and establish a rate of decay compatible with the
fractional diffusion limit.
