The Grazing Collision Limit of the Inelastic Kac Model around a Lévy-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., 14 (2004), pp. 1453–1480], when the equilibrium distribution function is a heavy-tailed Lévy-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.