Wave kinetic equation for Zakharov equation with stochastic dissipation and forcing

We develop a generalized wave kinetic theory for weakly nonlinear wave systems with stochastic dissipation. Starting from the Zakharov equation with multiplicative Gaussian noise, interpreted in the Itô sense, we derive the statistical evolution of the wave action using action–angle variables and a systematic multiscale expansion. Two distinct dissipation scalings are identified. For μ ∼ ϵ 2 , stochastic dissipation does not affect the leading–order kinetics and the classical Hasselmann equation is recovered. For μ ∼ ϵ , stochastic fluctuations survive averaging and induce an exponential modulation of resonant wave–wave interactions, together with a noise–induced drift term. For constant dissipation coefficients this reduces to a rescaling of the kinetic time, while in the genuinely stochastic regime temporal fluctuations reshape energy transfer in wave turbulence.