  In this paper, we consider an equation on random variables which can be
reduced to the equation which describes the evolution of systems of fermions.
We give some results of well-posedness for this equation on the spheres and
torus of dimension 2 and 3 and on the Euclidean space. We give results of
scattering and blow-up on the Euclidean depending on if the equation is
defocusing or focusing. We interpret the results in terms of the evolution of
fermions.
