Inelastic collapse of a randomly forced particle

We consider a randomly forced particle moving in a finite region, which rebounds inelastically with coefficient of restitution r on collision with the boundaries. We show that there is a transition at a critical value of r, r_c\equiv e^{-\pi/\sqrt{3}}, above which the dynamics is ergodic but beneath which the particle undergoes inelastic collapse, coming to rest after an infinite number of collisions in a finite time. The value of r_c is argued to be independent of the size of the region or the presence of a viscous damping term in the equation of motion.