  We show that for particles performing Brownian motion in a frozen array of
scatterers long-time correlations emerge in the mean-square displacement.
Defining the velocity autocorrelation function (VACF) via the second
time-derivative of the mean-square displacement, power-law tails govern the
long-time dynamics similar to the case of ballistic motion. The physical origin
of the persistent memory is due to repeated encounters with the same obstacle
which occurs naturally in Brownian dynamics without involving other scattering
centers. This observation suggests that in this case the VACF exhibits these
anomalies already at first order in the scattering density. Here we provide an
analytic solution for the dynamics of a tracer for a dilute planar Lorentz gas
and compare our results to computer simulations. Our result support the idea
that quenched disorder provides a generic mechanism for persistent correlations
irrespective of the microdynamics of the tracer particle.
