Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space
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| Format: | Preprint |
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2025
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| author | Pinzi, Alessandro Savaré, Giuseppe |
| author_facet | Pinzi, Alessandro Savaré, Giuseppe |
| contents | We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is $\mathbb{R}^d$, we study the differential structure of $\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))$ in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space Pinzi, Alessandro Savaré, Giuseppe Functional Analysis Metric Geometry Probability 49Q22, 35R06, 60G57, 53C23 We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is $\mathbb{R}^d$, we study the differential structure of $\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))$ in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems. |
| title | Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space |
| topic | Functional Analysis Metric Geometry Probability 49Q22, 35R06, 60G57, 53C23 |
| url | https://arxiv.org/abs/2510.07523 |