  We study the Boltzmann equation for a space-homogeneous gas of inelastic hard
spheres, with a diffusive term representing a random background forcing. Under
the assumption that the initial datum is a nonnegative $L^2$ function, with
bounded mass and kinetic energy (second moment), we prove the existence of a
solution to this model, which instantaneously becomes smooth and rapidly
decaying. Under a weak additional assumption of bounded third moment, the
solution is shown to be unique. We also establish the existence (but not
uniqueness) of a stationary solution. In addition we show that the
high-velocity tails of both the stationary and time-dependent particle
distribution functions are overpopulated with respect to the Maxwellian
distribution, as conjecturedby previous authors, and we prove pointwise lower
estimates for the solutions.
