GLOBAL WELL-POSEDNESS AND STABILITYOF THE INHOMOGENEOUS KINETICWAVE EQUATION NEAR VACUUM

. 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, for such dimensions, to the cubic part of the quantum Boltzmann equation for bosons, with Maxwell or hard potential and no collisional averaging.