Ultralimits of Wasserstein spaces and metric measure spaces with Ricci curvature bounded from below

Abstract

We investigate the stability of the Wasserstein distance, a metric structure on the space of probability measures arising from the theory of optimal transport, under metric ultralimits. We first show that if $(X_{i},d_{i})_{i\in\mathbb{N}}$ is a sequence of metric spaces with metric ultralimit $(\hat{X},\hat{d})$, then the p-Wasserstein space $(\mathcal{P}_{p}(\hat{X}),W_{p})$ embeds isometrically in a canonical fashion into the metric ultralimit of the sequence of p-Wasserstein spaces $(\mathcal{P}_{p}(X_{i}),W_{p})$. Second, using a notion of ultralimit of metric measure spaces modeled on the one introduced by Elek, we use the machinery of ultralimits of Wasserstein spaces to prove that an ultralimit of CD(K,$\infty$) spaces is a CD(K,$\infty$) space. This provides a new proof that the CD(K,$\infty$) property is stable under pointed measured Gromov convergence. Along the way, we establish some basic results on how the Loeb measure construction interacts with Wasserstein distances as well as integral functionals, which may be of independent interest.