  In this work, we consider non-linear corrections to the Langevin effective
theory of a heavy quark moving through a strongly coupled CFT plasma. In
AdS/CFT, this system can be identified with that of a string stretched between
the boundary and the horizon of an asymptotically AdS black-brane solution. We
compute the Feynman-Vernon influence phase for the heavy quark by evaluating
the Nambu-Goto action on a doubled string configuration. This configuration is
the linearised solution of the string motion in the doubled black-brane
geometry which has been proposed as the holographic dual of a thermal
Schwinger-Keldysh contour of the CFT. Our expression for the influence phase
passes non-trivial consistency conditions arising from the underlying unitarity
and thermality of the bath. The local effective theory obeys the recently
proposed non-linear fluctuation dissipation theorem relating the
non-Gaussianity of thermal noise to the thermal jitter in the damping constant.
This furnishes a non-trivial check for the validity of these relations derived
in the weak coupling regime.
