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Complete characterization of convergence to equilibrium for an inelastic Kac model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $α$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $α=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $α$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.