  We consider the spatially homogeneous Landau equation of kinetic theory, and
provide a differential inequality for the Wasserstein distance with quadratic
cost between two solutions. We deduce some well-posedness results. The main
difficulty is that this equation presents a singularity for small relative
velocities. Our uniqueness result is the first one in the important case of
soft potentials. Furthermore, it is almost optimal for a class of moderately
soft potentials, that is for a moderate singularity. Indeed, in such a case,
our result applies for initial conditions with finite mass, energy, and
entropy. For the other moderatley soft potentials, we assume additionnally some
moment conditions on the initial data. For very soft potentials, we obtain only
a local (in time) well-posedness result, under some integrability conditions.
Our proof is probabilistic, and uses a stochastic version of the Landau
equation, in the spirit of Tanaka.
