A limit theorem to a time-fractional diffusion

We prove a limit theorem for an integral functional of a Markov process. The Markovian dynamics is characterized by a linear Boltzmann equation modeling a one-dimensional test particle of mass $\lambda^{-1}\gg 1$ in an external periodic potential and undergoing collisions with a background gas of particles with mass one. The object of our limit theorem is the time integral of the force exerted on the test particle by the potential, and we consider this quantity in the limit that $\lambda$ tends to zero for time intervals on the scale $\lambda^{-1}$. Under appropriate rescaling, the total drift in momentum generated by the potential converges to a Brownian motion time-changed by the local time at zero of an Ornstein-Uhlenbeck process.