A study of the metric measure space of probability measures via a purely atomic superposition principle

We study the continuity equation on the metric measure space $({\mathcal{P}}_p(X),W_p,Q)$, when $X$ is either the Euclidean space or a compact, oriented, and boundaryless Riemannian manifold, for some suitable reference measure $Q\in {\mathcal{P}}_p({\mathcal{P}}_p(X))$, which by construction are concentrated over purely atomic measures. In fact, we consider the equation ${\mathrm{\partial }}_t{M}_t+{\mathrm{div}}_{\mathcal{P}}(b_t{M}_t)=0$, where $(M_t{)}_{t\in [0,T]}\subset \mathcal{P}(\mathcal{P}(X))$ and $b:[0,T]\times X\times \mathcal{P}(X)\to TX$, assuming that $M_t\ll Q$ for all $t\in [0,T]$, to then show when the purely atomic property is inherited by the liftings of the curve $M_t$ given by the nested superposition principle. On the Euclidean space, the main assumption is that the $r$-capacity of the diagonal $\mathrm{\Delta }\subset {\mathbb{R}}^d\times {\mathbb{R}}^d$ is zero with respect to $\nu \otimes \nu$, where $\nu$ is the barycenter of the reference measure $Q$. We will give sufficient conditions to ensure it, and in particular, thanks to the Nash embedding theorem, this analysis will allow us to extent the main results from the Euclidean space to Riemannian manifolds. Finally, we exploit this atomic superposition principle to show the lack of the Sobolev-to-Lipschitz property and the Poincaré inequality for functions in $W^{1,p}({\mathcal{P}}_p(X),W_p,Q)$. Then, we complete the analysis showing that, however, the $L^2$-Wasserstein space endowed with suitable reference measure $Q$, satisfies a Bakry–Émery curvature condition.