  We develop fluctuational electrodynamics for media with nonlinear optical
response in and out of thermal equilibrium. Starting from the stochastic
nonlinear Helmholtz equation and using the fluctuation dissipation theorem, we
obtain perturbatively a deterministic nonlinear Helmholtz equation for the
average field, the physical linear response, as well as the fluctuations and
Rytov currents. We show that the effects of nonlinear optics, in or out of
thermal equilibrium, can be taken into account with an effective, system-aware
dielectric function. We discuss the heat radiation of a planar, nonlinear
surface, showing that Kirchhoff's must be applied carefully. We find that the
spectral emissivity of a nonlinear nanosphere can in principle be negative,
implying the possibility of heat flow reversal for specific frequencies.
