Random Surfaces and Higher Algebra
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914301985423360 |
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| author | Lee, Darrick Oberhauser, Harald |
| author_facet | Lee, Darrick Oberhauser, Harald |
| contents | We introduce a characteristic function for laws of random surfaces $\mathbf{X}: [0,s] \times [0,t] \to \mathbb{R}^d$, in the spirit of expected path developments for one-dimensional stochastic processes into matrix groups. A key property is that path development is structure preserving: path concatenation becomes matrix multiplication. The main challenge is to account for two distinct concatenation operations for surfaces: horizontal and vertical. To address this, we use the notion of surface holonomy from higher geometry to define surface developments, and study this in a stochastic context. We generalize surface developments to the Young setting of $ρ$-Hölder surfaces, where $ρ> \frac12$, show that such developments characterize parametrized surfaces. Our main result shows that the resulting expected surface development provides a computable and structured description of laws of random surfaces and leads to a natural metric on the space of probability measures on surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08366 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random Surfaces and Higher Algebra Lee, Darrick Oberhauser, Harald Probability Algebraic Topology Category Theory Differential Geometry We introduce a characteristic function for laws of random surfaces $\mathbf{X}: [0,s] \times [0,t] \to \mathbb{R}^d$, in the spirit of expected path developments for one-dimensional stochastic processes into matrix groups. A key property is that path development is structure preserving: path concatenation becomes matrix multiplication. The main challenge is to account for two distinct concatenation operations for surfaces: horizontal and vertical. To address this, we use the notion of surface holonomy from higher geometry to define surface developments, and study this in a stochastic context. We generalize surface developments to the Young setting of $ρ$-Hölder surfaces, where $ρ> \frac12$, show that such developments characterize parametrized surfaces. Our main result shows that the resulting expected surface development provides a computable and structured description of laws of random surfaces and leads to a natural metric on the space of probability measures on surfaces. |
| title | Random Surfaces and Higher Algebra |
| topic | Probability Algebraic Topology Category Theory Differential Geometry |
| url | https://arxiv.org/abs/2311.08366 |