We will talk about global dispersive solutions to the space inhomogeneous kinetic wave equation (KWE) which propagate $L^1_{xv}$ -- moments and conserve mass, momentum and energy. We prove that they scatter, and that the wave operators mapping the initial data to the scattering states are 1-1, onto and continuous in a suitable topology.

This is the first global existence result for strong solutions for KWE. This contrasts with prior global existence results for mild solutions, which satisfy a transported version of the equation but do not solve the equation itself.

Our proof is carried out entirely in physical space and combines dispersive estimates for the free transport with new trilinear bounds for the gain and loss operators of the KWE on weighted Lebesgue spaces. The main difficulty is the fast growth of the hard-sphere kernel. Our fundamental tool to handle it is a novel collisional averaging estimate.

We also show that the nonlinear evolution preserves positivity forward in time. For this, we use the Kaniel-Shinbrot iteration scheme, properly initialized to ensure successive approximations are dispersive.