Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918066332368896 |
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| author | Papageorgiou, Effie Trojan, Bartosz |
| author_facet | Papageorgiou, Effie Trojan, Bartosz |
| contents | We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17042 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings Papageorgiou, Effie Trojan, Bartosz Functional Analysis Primary: 35K08, 35B40, 51E24, 58J35, 60B15. Secondary: 20E42, 20F55, 22E35, 43A90, 60G50 We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion. |
| title | Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings |
| topic | Functional Analysis Primary: 35K08, 35B40, 51E24, 58J35, 60B15. Secondary: 20E42, 20F55, 22E35, 43A90, 60G50 |
| url | https://arxiv.org/abs/2506.17042 |