  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.
