Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media

We consider a space-homogeneous gas of {it inelastic hard spheres}, with a {it diffusive term} representing a random background forcing (in the framework of so-called {em constant normal restitution coefficients} $alpha in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $alpha in [alpha_*,1)$ for some constructive $alpha_* in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $alpha in [alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.