Ester Gabetta, Eugenio Regazzini
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 $\alpha$ is determined by the so-called degree of
inelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. 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 $\alpha$. (2) To determine
the solution of a well-known stochastic functional equation in the absence of
extra-conditions usually adopted.
View original:
http://arxiv.org/abs/1202.3633
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