  The sliced-Wasserstein flow is an evolution equation where a probability
density evolves in time, advected by a velocity field computed as the average
among directions in the unit sphere of the optimal transport displacements from
its 1D projections to the projections of a fixed target measure. This flow
happens to be the gradient flow in the usual Wasserstein space of the squared
sliced-Wasserstein distance to the target. We consider the question whether in
long-time the flow converges to the target (providing a positive result when
the target is Gaussian) and the question of the long-time limit of the flow map
obtained by following the trajectories of each particle. We prove that this
limit is in general not the optimal transport map from the starting measure to
the target. Both questions come from the folklore about sliced-Wasserstein and
had never been properly treated.
