I will discuss asymptotic estimates for the p− Wasserstein distance Wp on the set of probability measures. In particular, the limit

Dμ,q(ν):=limϵ→0ϵ−qWp(μ+ϵν,μ),

where μ is a probability measure and ν a signed, neutral measure. It will be shown that Dμ,q with q=1 is a finite norm on measures supported within the support of μ, provided ∫ϕdν can be controlled in terms of ∫∣∣∇ϕ∣∣p/(p−1)dμ, for any smooth function ϕ. As an application I obtain the existence of a velocity field driving a flow of probability measures which is absolutely continuous in the total variation norm.

I will also discuss the case where the limit Dμ,p with q=1 is not finite. It will be shown that the limit is a finite norm on some reduced space, provided the exponent q∈[1/p,1) is given in terms of p and the dimension d(μ) (to be defined). The case where the support of μ is disconnected (corresponding to d(μ)=∞)) corresponds to q=1/p and is called "optimal teleportation". I will show that in this case the limit Dμ,1/p is reduced to a norm on a graph whose vertices correspond to the connected components of he support of μ.