  The sub-arcsec bright points (BP) associated with the small scale magnetic
fields in the lower solar atmosphere are advected by the evolution of the
photospheric granules. We measure various quantities related to the horizontal
motions of the BPs observed in two wavelengths, including the velocity
auto-correlation function. A 1 hr time sequence of wideband H$\alpha$
observations conducted at the \textit{Swedish 1-m Solar Telescope}
(\textit{SST}), and a 4 hr \textit{Hinode} \textit{G}-band time sequence
observed with the Solar Optical telescope are used in this work. We follow 97
\textit{SST} and 212 \textit{Hinode} BPs with 3800 and 1950 individual velocity
measurements respectively. For its high cadence of 5 s as compared to 30 s for
\textit{Hinode} data, we emphasize more on the results from \textit{SST} data.
The BP positional uncertainty achieved by \textit{SST} is as low as 3 km. The
position errors contribute 0.75 km$^2$ s$^{-2}$ to the variance of the observed
velocities. The \textit{raw} and \textit{corrected} velocity measurements in
both directions, i.e., $(v_x,v_y)$, have Gaussian distributions with standard
deviations of $(1.32,1.22)$ and $(1.00, 0.86)$ km s$^{-1}$ respectively. The BP
motions have correlation times of about $22 - 30$ s. We construct the power
spectrum of the horizontal motions as a function of frequency, a quantity that
is useful and relevant to the studies of generation of Alfv\'en waves.
Photospheric turbulent diffusion at time scales less than 200 s is found to
satisfy a power law with an index of 1.59.
