Fluctuations in a kinetic transport model for quantum friction

We study a linear Boltzmann equation describing the effective dynamics of a quantum particle moving through an ideal Bose gas at zero temperature. The particle emits sound waves into the Bose gas, a process causing it to slow down. This mechanism for friction is akin to one exhibited by a charged particle travelling through an optically dense medium at a speed larger than the speed of light and hence emitting Čerenkov radiation. We study the spatial trajectory of the Markov process corresponding to the Boltzmann equation and prove that this process converges to a multiple of Brownian motion after an exponential rescaling of time. We also show that the asymptotic position of the particle remains finite on average, but that its absolute value diverges logarithmically. Our unusual exponential rescaling of time is appropriate for a description of friction at zero temperature and is distinct from diffusive scaling appropriate at positive temperature. It is also shown that if random fluctuations of the particle position were neglected the latter would turn out to diverge.