I will talk about a one-dimensional model for a massive particle immersed in a gas of light particles and under the influence of a microscopic periodic potential. The starting point is a Markovian description of the massive test particle dynamics given by a linear Boltzmann equation. In the Brownian limit, the influence of the microscopic potential vanishes, and the rescaled momentum of the particle converges to an Ornstein-Uhlenbeck process. On a smaller scale, the total drift in momentum due to the forcing of the potential converges to a Brownian motion time-changed by the local time at zero of the limiting OrnsteinUhlenbeck process. Our result is related to the literature on limit theorems for additive functionals of null recurrent Markov processes. This is joint work with Loc Dubois.