Expected Signature on a Riemannian Manifold and Its Geometric Implications
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866917725643735040 |
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| author | Geng, Xi Ni, Hao Wang, Chaorui |
| author_facet | Geng, Xi Ni, Hao Wang, Chaorui |
| contents | On a compact Riemannian manifold $M,$ we show that the Riemannian distance function $d(x,y)$ can be explicitly reconstructed from suitable asymptotics of the expected signature of Brownian bridge from $x$ to $y$. In addition, by looking into the asymptotic expansion of the fourth level expected signature of the Brownian loop based at $x\in M$, one can explicitly reconstruct both intrinsic (Ricci curvature) and extrinsic (second fundamental form) curvature properties of $M$ at $x$. As independent interest, we also derive the intrinsic PDE for the expected Brownian signature dynamics on $M$ from the perspective of the Eells-Elworthy-Malliavin horizontal lifting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expected Signature on a Riemannian Manifold and Its Geometric Implications Geng, Xi Ni, Hao Wang, Chaorui Probability On a compact Riemannian manifold $M,$ we show that the Riemannian distance function $d(x,y)$ can be explicitly reconstructed from suitable asymptotics of the expected signature of Brownian bridge from $x$ to $y$. In addition, by looking into the asymptotic expansion of the fourth level expected signature of the Brownian loop based at $x\in M$, one can explicitly reconstruct both intrinsic (Ricci curvature) and extrinsic (second fundamental form) curvature properties of $M$ at $x$. As independent interest, we also derive the intrinsic PDE for the expected Brownian signature dynamics on $M$ from the perspective of the Eells-Elworthy-Malliavin horizontal lifting. |
| title | Expected Signature on a Riemannian Manifold and Its Geometric Implications |
| topic | Probability |
| url | https://arxiv.org/abs/2407.13086 |