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 Alfven waves. Photospheric turbulent diffusion at time scales less than 200 s is found to satisfy a power law with an index of 1.59.
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