  We evaluate the Brownian motion of a massive particle ("black hole") at the
center of a galaxy using N-body simulations. Our galaxy models have power-law
central density cusps like those observed at the centers of elliptical
galaxies. The simulations show that the black hole achieves a steady-state
kinetic energy that is substantially different than would be predicted based on
the properties of the galaxy model in the absence of the black hole. The reason
appears to be that the black hole responds to stars whose velocities have
themselves been raised by the presence of the black hole. Over a wide range of
density slopes and black hole masses, the black hole's mean kinetic energy is
equal to what would be predicted under the assumption that it is in energy
equipartition with stars lying within a distance ~r_h/2 from it, where r_h is
the black hole's influence radius. The dependence of the Brownian velocity on
black hole mass is approximately <V^2>~ 1/M^{1/(3-gamma)} with gamma the
power-law index of the stellar density profile, rho~1/r^gamma. This is less
steep than the 1/M dependence predicted in a model where the effect of the
black hole on the stellar velocities is ignored. The influence of a stellar
mass spectrum on the black hole's Brownian motion is also evaluated and found
to be consistent with predictions from Chandrasekhar's theory. We use these
results to derive a probability function for the mass of the Milky Way black
hole based on a measurement of its proper motion velocity. Interesting
constraints on M will require a velocity resolution exceeding 0.5 km/s.
