  In this paper we devote our attention to a class of weighted ultrafast
diffusion equations arising from the problem of quantisation for probability
measures. These equations have a natural gradient flow structure in the space
of probability measures endowed with the quadratic Wasserstein distance.
Exploiting this structure, in particular through the so-called JKO scheme, we
introduce a notion of weak solutions, prove existence, uniqueness, BV and H^1
estimates, L^1 weighted contractivity, Harnack inequalities, and exponential
convergence to a steady state.
