  We show that anomalous diffusion arises in two different models for the
motion of randomly forced and weakly damped particles: one is a generalisation
of the Ornstein-Uhlenbeck process with a random force which depends on position
as well as time, the other is a generalisation of the Chandrasekhar-Rosenbluth
model of stellar dynamics, encompassing non-Coulombic potentials. We show that
both models exhibit anomalous diffusion of position $x$ and momentum $p$ with
the same exponents: $<x^2> \sim C_x t^2$ and $<p^2> \sim C_p t^{2/5}$. We are
able to determine the prefactors $C_x$, $C_p$ analytically.
