Continuous transformations of probability measures and their transport representations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918453095432192 |
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| author | Lavenant, Hugo Savaré, Giuseppe |
| author_facet | Lavenant, Hugo Savaré, Giuseppe |
| contents | Given a function $F$ transforming a probability measure $μ$ into another one $F(μ)$, we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image $F(μ)$ of the input probability measure $μ$ as the push-forward of $μ$ by a map $f(\cdot,μ)$ which may depend on $μ$; and furthermore, how regular $f$ can be chosen depending on $F$. Even if $F$ is continuous and a transport representative exists, it cannot necessarily be chosen in a continuous way; however, if $F$ is Lipschitz continuous with respect to the Wasserstein distance, then $f$ can be chosen continuous. We provide several examples to illustrate the sharpness of our assumptions. This question is motivated by approximation results for transformations of probability distributions with transformers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16653 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuous transformations of probability measures and their transport representations Lavenant, Hugo Savaré, Giuseppe Functional Analysis Optimization and Control Probability 49Q22, 28A33, 28D05, 60B05, 41A65 Given a function $F$ transforming a probability measure $μ$ into another one $F(μ)$, we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image $F(μ)$ of the input probability measure $μ$ as the push-forward of $μ$ by a map $f(\cdot,μ)$ which may depend on $μ$; and furthermore, how regular $f$ can be chosen depending on $F$. Even if $F$ is continuous and a transport representative exists, it cannot necessarily be chosen in a continuous way; however, if $F$ is Lipschitz continuous with respect to the Wasserstein distance, then $f$ can be chosen continuous. We provide several examples to illustrate the sharpness of our assumptions. This question is motivated by approximation results for transformations of probability distributions with transformers. |
| title | Continuous transformations of probability measures and their transport representations |
| topic | Functional Analysis Optimization and Control Probability 49Q22, 28A33, 28D05, 60B05, 41A65 |
| url | https://arxiv.org/abs/2604.16653 |