  We provide a rigorous justification of various kinetic regimes exhibited by
the nonlinear Schr\"{o}dinger equation with an additive stochastic forcing and
a viscous dissipation. The importance of such damped-driven models stems from
their wide empirical use in studying turbulence for nonlinear wave systems. The
force injects energy into the system at large scales, which is then transferred
across scales, thanks to the nonlinear wave interactions, until it is
eventually dissipated at smaller scales. The presence of such scale-separated
forcing and dissipation allows for the constant flux of energy in the
intermediate scales, known as the inertial range, which is the focus of the
vast amount of numerical and physical literature on wave turbulence.
  Roughly speaking, our results provide a rigorous kinetic framework for this
turbulent behavior by proving that the stochastic dynamics can be effectively
described by a deterministic damped-driven kinetic equation, which carries the
full picture of the turbulent energy dynamic across scales (like cascade
spectra or other flux solutions). The analysis extends previous works in the
unperturbed setting arXiv:1912.09518-arXiv:2301.07063 to the above empirically
motivated damped driven setting. Here, in addition to the size $L$ of the
system, and the strength $\lambda$ of the nonlinearity, an extra thermodynamic
parameter has to be included in the kinetic limit ($L\to \infty, \lambda\to
0$), namely the strength $\nu$ of the forcing and dissipation. Various regimes
emerge depending on the relative sizes of $L$, $\lambda$ and $\nu$, which give
rise to different kinetic equations. Two major novelties of this work is the
extension of the Feynman diagram analysis to additive stochastic objects, and
the sharp asymptotic development of the leading terms in that expansion.
