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 reflection satisfy \(v_f=-rv_i\), 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 \(P(x,v)\).