Random Walks on Virtual Persistence Diagrams
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918408874885120 |
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| author | Fanning, Charles Aktas, Mehmet |
| author_facet | Fanning, Charles Aktas, Mehmet |
| contents | In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent $λ_H$ satisfying $λ_H(θ)=\sum_{κ\in H\setminus\{0\}}\bigl(1-\Reθ(κ)\bigr)ν(κ),$ for a symmetric $ν\in\ell^1(H\setminus\{0\})$. This gives a symmetric jump process on $H$. The exponent $λ_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02117 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random Walks on Virtual Persistence Diagrams Fanning, Charles Aktas, Mehmet Probability Algebraic Topology Functional Analysis 60J27, 60B15, 43A35, 46E22, 31C20, 55N31 In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent $λ_H$ satisfying $λ_H(θ)=\sum_{κ\in H\setminus\{0\}}\bigl(1-\Reθ(κ)\bigr)ν(κ),$ for a symmetric $ν\in\ell^1(H\setminus\{0\})$. This gives a symmetric jump process on $H$. The exponent $λ_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics. |
| title | Random Walks on Virtual Persistence Diagrams |
| topic | Probability Algebraic Topology Functional Analysis 60J27, 60B15, 43A35, 46E22, 31C20, 55N31 |
| url | https://arxiv.org/abs/2603.02117 |