  We study Brownian motion and the associated Langevin equation in AdS/CFT. The
Brownian particle is realized in the bulk spacetime as a probe fundamental
string in an asymptotically AdS black hole background, stretching between the
AdS boundary and the horizon. The modes on the string are excited by the
thermal black hole environment and consequently the string endpoint at the
boundary undergoes an erratic motion, which is identified with an external
quark in the boundary CFT exhibiting Brownian motion. Semiclassically, the
modes on the string are thermally excited due to Hawking radiation, which
translates into the random force appearing in the boundary Langevin equation,
while the friction in the Langevin equation corresponds to the excitation on
the string being absorbed by the black hole. We give a bulk proof of the
fluctuation-dissipation theorem relating the random force and friction. This
work can be regarded as a step toward understanding the quantum microphysics
underlying the fluid-gravity correspondence. We also initiate a study of the
properties of the effective membrane or stretched horizon picture of black
holes using our bulk description of Brownian motion.
