  It has long been known that weakly nonlinear field theories can have a
late-time stationary state that is not the thermal state, but a wave turbulent
state with a far-from-equilibrium cascade of energy. We go beyond the existence
of the wave turbulent state, studying fluctuations about the wave turbulent
state. Specifically, we take a classical field theory with an arbitrary quartic
interaction and add dissipation and Gaussian-random forcing. Employing the path
integral relation between stochastic classical field theories and quantum field
theories, we give a prescription, in terms of Feynman diagrams, for computing
correlation functions in this system. We explicitly compute the two-point and
four-point functions of the field to next-to-leading order in the coupling.
Through an appropriate choice of forcing and dissipation, these correspond to
correlation functions in the wave turbulent state. In particular, we derive the
kinetic equation to next-to-leading order.
