  We analyze a system of stochastic differential equations describing the joint
motion of a massive (inert) particle in a viscous fluid in the presence of a
gravitational field and a Brownian particle impinging on it from below, which
transfers momentum proportional to the local time of collisions. We study the
long-time fluctuations of the velocity of the inert particle and the gap
between the two particles, and we show convergence in total variation to the
stationary distribution is exponentially fast. We also produce matching upper
and lower bounds on the tails of the stationary distribution and show how these
bounds depend on the system parameters. A renewal structure for the process is
established, which is the key technical tool in proving the mentioned results.
