  We study the mathematical properties of a kinetic equation which describes
the long time behaviour of solutions to the weak turbulence equation associated
to the cubic nonlinear Schr\"odinger equation. In particular, we give a precise
definition of weak solutions and prove global existence of solutions for all
initial data with finite mass. We also prove that any nontrivial initial datum
yields the instantaneous onset of a condensate, i.e. a Dirac mass at the origin
for any positive time. Furthermore we show that the only stationary solutions
with finite total measure are Dirac masses at the origin. We finally construct
solutions with finite energy, which is transferred to infinity in a
self-similar manner.
