  In this paper, we prove global in time existence, uniqueness and stability of
mild solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation,
for Laplacian dispersion relation in dimension $d=2,3$. We also show that for
non-negative initial data, the solution remains non-negative. This is achieved
by connecting the inhomogeneous kinetic wave equation to the cubic part of a
quantum Boltzmann-type equation with moderately hard potential and no
collisional averaging.
