A study of the metric measure space of probability measures via a purely atomic superposition principle
Alessandro Pinzi
TL;DR
The paper develops a framework for the continuity equation on the metric measure space of probability measures endowed with a reference measure supported on purely atomic distributions. It introduces a nested atomic superposition principle, enables lifting curves of atomic measures to atomic curve laws under capacitary conditions on the diagonal, and extends these results to compact Riemannian manifolds via Nash embedding. It then reveals a lack of Sobolev-to-Lipschitz and Poincaré inequalities in the associated $L^2$-Wasserstein setting, while establishing Bakry–Émery curvature properties for the Wasserstein space under suitable hypotheses. Taken together, these results provide a detailed atomic-measures perspective on dynamical evolutions in Wasserstein spaces and connect probabilistic, geometric, and functional-analytic aspects of metric-measure spaces.
Abstract
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 $\partial_t M_t +\operatorname{div}_{\mathcal{P}}(b_t M_t) = 0$, 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 $Δ\subset \mathbb{R}^d \times \mathbb{R}^d$ is zero with respect to $ν\otimes ν$, where $ν$ 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.