Homogeneous kinetic equations model the redistribution of energy and momentum in an
ensemble of particles which interact in binary collisions. In the original Boltzmann
equation, these collisions are fully elastic, i.e., they conserve the total energy
and momentum of the two interacting particles precisely. It is then a consequence of
Boltzmann's celebrated H-theorem that the ensemble's velocity distribution
eventually becomes Gaussian.

Over the last decade, inelastic models of various kinds have been introduced. In
those, collisions are not fully elastic anymore, e.g. due to interactions with a
background heat bath. This leads to a statistical gain and/or loss of the total
energy in binary collisions. The stationary velocity distributions become much more
interesting: they are stochastic mixtures of stable laws and thus possess fat energy
tails.

In this talk, we present a probabilistic approach to study the long time behavior in
a class of inelastic kinetic models. Our key tools are the classical central limit
theorem and a special representation of the Wild sum, which has been introduced
recently by Dolera and Regazzini.

This is joint work with Federico Bassetti (UNIPV) and Lucia Ladelli (POLIMI).