systems far from equilibrium no such claims can be made. Although some similarities can exist [@sieberer_2016_keldysh; @fossfeig_optical_bistability; @Jose_Iacopo], driving and dissipation in general prevent the system from approaching a complete thermal equilibrium [@carmichael_statistical]. As a result, the phase-space distribution of out-of-equilibrium systems is typically much more sensitive to the actual microscopic driving, dissipation and equilibration processes that drive the system towards the steady-state.

In this manuscript we study a weakly interacting driven-dissipative Bose-Hubbard model in the superfluid regime. Going beyond the widely applied Bogoliubov approximation, which assumes noninteracting quasiparticles, we show that scattering processes involving three quasiparticles, known as Beliaev-Landau scattering [@beliaev1958], leave a small imprint on the steady-state momentum distribution and are responsible for non-Gaussian features in the photon statistics of the cavity output field.

Beliaev-Landau scattering processes were originally predicted in the many-body theory of quantum fluids [@PinesNozieres] to be responsible for the finite lifetime of phonons in systems of either bosonic [@Pitaevskii_Stringari_BL; @Giorgini_BL; @Domokos_beliaev] or fermionic particles [@castin_landau] and were experimentally observed with ultracold atomic gases[@Foot_BL; @Steinhauer_beliaev]. In these works, the damping was typically detected after a population of phonons was introduced in the system by an external perturbation. Here, on the contrary, we aim at capturing 