  We study the Brownian dynamics of a solid particle on a vibrating solid
surface. Phenomenologically, the interaction between the two solid surfaces is
modeled by solid friction, and the Gaussian white noise models the vibration of
the solid surface. The solid friction is proportional to the sign of relative
velocity. We derive the Fokker-Planck (FP) equation for the time-dependent
probability distribution to find the particle at a given location. We calculate
analytically the steady state velocity distribution function, mean-squared
velocity and diffusion coefficient in $d$ dimensions. We present a generic
method of calculating the autocorrelations in $d$ dimensions. This results in
one dimension in an exact evaluation of the steady state velocity
autocorrelation. In higher dimensions, our exact general expression enables the
analytic evaluation of the autocorrelation to any required approximation. We
present approximate analytic expressions in two and three dimensions. Next, we
numerically calculate the mean-square velocity and steady state velocity
autocorrelation function up to $d=3$. Our numerical results are in good
agreement with the analytically obtained results.
