  We consider a linear Boltzmann equation that arises in a model for quantum
friction. It describes a particle that is slowed down by the emission of
bosons. We study the stochastic process generated by this Boltzmann equation
and we show convergence of its spatial trajectory to a multiple of Brownian
motion with exponential scaling. The asymptotic position of the particle is
finite in mean, even though its absolute value is typically infinite. This is
contrasted to an approximation that neglects the influence of fluctuations,
where the mean asymptotic position is infinite.
