The grazing collision limit of the inelastic Kac model around a   L\'evy-type equilibrium
  This paper is devoted to the grazing collision limit of the inelastic Kac model introduced in [A. Pulvirenti and G. Toscani. J. Statist. Phys., 114(5-6):1453--1480, 2004], when the equilibrium distribution function is a heavy-tailed L\'evy-type distribution with infinite variance. We prove that solutions in an appropriate domain of attraction of the equilibrium distribution converge to solutions of a Fokker-Planck equation with a fractional diffusion operator. 
G. Furioli, A. Pulvirenti, E. Terraneo and G. Toscani
math.AP