  We obtain a fluctuation--dissipation theorem describing thermal
electromagnetic fluctuation effects in nonlinear media that we exploit in
conjunction with a stochastic Langevin framework to study thermal radiation
from Kerr ($\chi^{(3)}$) photonic cavities coupled to external environments at
and out of equilibrium. We show that that in addition to thermal broadening due
to two-photon absorption,the emissivity of such cavities can exhibit
asymmetric,non-Lorentzian lineshapes due to self-phase modulation. When the
local temperature of the cavity is larger than that of the external bath, we
find that the heat transfer into the bath exceeds the radiation from a
corresponding linear black body at the same local temperature. We predict that
these temperature-tunable thermal processes can be observed in practical,
nanophotonic cavities operating at relatively small temperatures.
