Localisation for constrained transports I: theory
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arXiv
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| Format: | Preprint |
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2023
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| author | Ciosmak, Krzysztof J. |
| author_facet | Ciosmak, Krzysztof J. |
| contents | We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures $μ,ν$ ordered with respect to a cone $\mathcal{F}$ of functions on $Ω$ stable under maxima. Under the assumption that any $\mathcal{F}$-transport between $μ$ and $ν$ is local, we establish the existence of the finest partitioning of $Ω$, depending only on $μ,ν$ and the cone $\mathcal{F}$, into $\mathcal{F}$-convex sets, called irreducible components, such that any $\mathcal{F}$-transport between $μ$ and $ν$ must adhere to this partitioning.
Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all $\mathcal{F}$-transports between $μ$ and $ν$ if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting.
Among our contributions is also a generalisation of the Strassen's theorem to the setting of generalised convexity |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12281 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Localisation for constrained transports I: theory Ciosmak, Krzysztof J. Probability Functional Analysis Primary: 49N05, 49Q22, 60D05, 60G42, 60G48, Secondary: 06B23, 28A50, 46E05 We investigate an analogue of the irreducible convex paving in the context of generalised convexity. Consider two Radon probability measures $μ,ν$ ordered with respect to a cone $\mathcal{F}$ of functions on $Ω$ stable under maxima. Under the assumption that any $\mathcal{F}$-transport between $μ$ and $ν$ is local, we establish the existence of the finest partitioning of $Ω$, depending only on $μ,ν$ and the cone $\mathcal{F}$, into $\mathcal{F}$-convex sets, called irreducible components, such that any $\mathcal{F}$-transport between $μ$ and $ν$ must adhere to this partitioning. Furthermore, we demonstrate that a set, whose sections are contained in the corresponding irreducible components, is a polar set with respect to all $\mathcal{F}$-transports between $μ$ and $ν$ if and only if it is a polar set with respect to all transports. This provides an affirmative answer to a generalisation of a conjecture proposed by Obłój and Siorpaes regarding polar sets in the martingale transport setting. Among our contributions is also a generalisation of the Strassen's theorem to the setting of generalised convexity |
| title | Localisation for constrained transports I: theory |
| topic | Probability Functional Analysis Primary: 49N05, 49Q22, 60D05, 60G42, 60G48, Secondary: 06B23, 28A50, 46E05 |
| url | https://arxiv.org/abs/2312.12281 |