The Monge-Kantorovich problem on Wasserstein space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913536838467584 |
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| author | Emami, Pedram Pass, Brendan |
| author_facet | Emami, Pedram Pass, Brendan |
| contents | We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures $\mathbb{P}_1,\mathbb{P}_2\in\mathcal{P}(\mathcal{P}(M))$ on the space $\mathcal{P}(M)$ of probability measures on a smooth compact manifold, we study the optimal transport problem between $\mathbb{P}_1$ and $\mathbb{P}_2 $ where the cost function is given by the squared Wasserstein distance $W_2^2(μ,ν)$ between $μ,ν\in \mathcal{P}(M)$. Under appropriate assumptions on $\mathbb{P}_1$, we prove that there exists a unique optimal plan and that it takes the form of an optimal map. An extension of this result to cost functions of the form $h(W_2(μ,ν))$, for strictly convex and strictly increasing functions $h$, is also established. The proofs rely heavily on a recent result of Schiavo \cite{schiavo2020rademacher}, which establishes a version of Rademacher's theorem on Wasserstein space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08585 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Monge-Kantorovich problem on Wasserstein space Emami, Pedram Pass, Brendan Probability Optimization and Control We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures $\mathbb{P}_1,\mathbb{P}_2\in\mathcal{P}(\mathcal{P}(M))$ on the space $\mathcal{P}(M)$ of probability measures on a smooth compact manifold, we study the optimal transport problem between $\mathbb{P}_1$ and $\mathbb{P}_2 $ where the cost function is given by the squared Wasserstein distance $W_2^2(μ,ν)$ between $μ,ν\in \mathcal{P}(M)$. Under appropriate assumptions on $\mathbb{P}_1$, we prove that there exists a unique optimal plan and that it takes the form of an optimal map. An extension of this result to cost functions of the form $h(W_2(μ,ν))$, for strictly convex and strictly increasing functions $h$, is also established. The proofs rely heavily on a recent result of Schiavo \cite{schiavo2020rademacher}, which establishes a version of Rademacher's theorem on Wasserstein space. |
| title | The Monge-Kantorovich problem on Wasserstein space |
| topic | Probability Optimization and Control |
| url | https://arxiv.org/abs/2406.08585 |