We give a simple recipe for computing dissipation and fluctuations
(commutator and anti-commutator correlation functions) for non-equilibrium
black hole geometries. The recipe formulates Hawking radiation as an initial
value problem, and is suitable for numerical work. We show how to package the
fluctuation and dissipation near the event horizon into correlators on the
stretched horizon. These horizon correlators determine the bulk and boundary
field theory correlation functions. In addition, the horizon correlators are
the components of a horizon effective action which provides a quantum
generalization of the membrane paradigm. In equilibrium, the analysis
reproduces previous results on the Brownian motion of a heavy quark. Out of
equilibrium, Wigner transforms of commutator and anti-commutator correlation
functions obey a fluctuation-dissipation relation at high frequency.Comment: 28 pages, 6 figure
