  We consider a generic Hamiltonian system of nonlinear interacting waves with
3-wave interactions. In the kinetic regime of wave turbulence, which assumes
weak nonlinearity and large system size, the relevant observable associated
with the wave amplitude is the empirical spectral density that appears as the
natural precursor of the spectral density, or spectrum, for finite system size.
Following classical derivations of the Peierls equation for the moment
generating function of the wave amplitudes in the kinetic regime, we propose a
large deviation estimate for the dynamics of the empirical spectral density,
where the number of admissible wavenumbers, which is proportional to the volume
of the system, appears as the natural large deviation parameter. The large
deviation stochastic Hamiltonian that quantifies the minus of the log
probability of a trajectory is computed within the kinetic regime which assumes
the Random Phase approximation for weak nonlinearity. We compare this
Hamiltonian with the one for a system of modes interacting in a mean-field way
with the empirical spectrum. Its relationship with the Random Phase and
Amplitude approximation is discussed. Moreover, for the specific case when no
forces and dissipation are present, a few fundamental properties of the large
deviation dynamics are investigated. We show that the latter conserves total
energy and momentum, as expected for a 3-wave interacting systems. In addition,
we compute the equilibrium quasipotential and check that global detailed
balance is satisfied at the large deviation level. Finally, we discuss briefly
some physical applications of the theory.
