  In this work, the motion of a dust particle under the influence of the random
force due to dust charge fluctuations is considered as a non-Markovian
stochastic process. Memory effects in the velocity process of the dust particle
are studied. A model is developed based on the fractional Langevin equation for
the motion of the dust grain. The fluctuation-dissipation theorem for the dust
grain is derived from this equation. The mean-square displacement and the
velocity autocorrelation function of the dust particle are obtained in terms of
the Mittag-Leffler functions. Their asymptotic behavior and the dust particle
temperature due to charge fluctuations are studied in the long-time limit. As
an interesting result, it is found that the presence of memory effects in the
velocity process of the dust particle as a non-Markovian process can cause an
anomalous diffusion in dusty plasmas. In this case, the velocity
autocorrelation function of the dust particle has a power-law decay like
$t^{-\alpha-2}$, where the exponent $\alpha$ take values $0<\alpha<1$.
