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Pulvirenti and Toscani introduced an equation which extends the Kaccaricature of a Maxwellian gas to inelastic particles. We show that theprobability distribution, solution of the relative Cauchy problem, convergesweakly to a probability distribution if and only if the symmetrized initialdistribution belongs to the standard domain of attraction of a symmetric stablelaw, whose index $\alpha$ is determined by the so-called degree ofinelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. This result isthen used: (1) To state that the class of all stationary solutions coincideswith that of all symmetric stable laws with index $\alpha$. (2) To determinethe solution of a well-known stochastic functional equation in the absence ofextra-conditions usually adopted.

Complete characterization of convergence to equilibrium for an inelastic Kac model