We consider the four waves spatial homogeneous kinetic equation arising in
wave turbulence theory. We study the long-time behaviour and existence of
solutions around the Rayleigh-Jeans equilibrium solutions. For cut-off’d
frequencies, we show that for dispersion relations weakly perturbed around the
quadratic case, the linearized operator around the Rayleigh-Jeans equilibria is
coercive. We then pass to the fully nonlinear operator, showing an $L^2$ –
stability for initial data close to Rayleigh-Jeans.