Equilibrium statistics of an inelastically bouncing ball, subject to gravity and a random force

We consider a particle moving on the half line x > 0 and subject to a constant force in the -x direction plus a delta-correlated random force. At x = 0 the particle is reflected inelastically. The velocities just after and before the reflection satisfy v(f) = -r v(i), where r is the coefficient of restitution. This simple model is of interest in connection with studies of driven granular matter in a gravitational field. With an exact analytical approach and simulations we study the steady-state distribution function P(x,v).