  We study the behavior of a tracer particle driven by a one-dimensional
fluctuating potential, defined initially as a Brownian motion, and evolving in
time according to the heat equation. We obtain two main results. First, in the
short time limit, we show that the fluctuations of the particle become Gaussian
and sub-diffusive, with dynamical exponent $3/4$. Second, in the long time
limit, we show that the particle is trapped by the local minima of the
potential and evolves diffusively i.e. with exponent $1/2$.
