Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910049639596032 |
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| author | Papageorgiou, Effie |
| author_facet | Papageorgiou, Effie |
| contents | We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to\infty$. Second, using them, we show that solutions with initial data in weighted $\ell^1$ classes, asymptotically factorize in $\ell^p$ norms, $p\in[1,\infty]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11232 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees Papageorgiou, Effie Analysis of PDEs Functional Analysis 35K08, 35B40, 43A85, 43A90, 22E35 We study the large-time behavior of the continuous-time heat kernel and of solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic formulas for the heat kernel as $t\to\infty$. Second, using them, we show that solutions with initial data in weighted $\ell^1$ classes, asymptotically factorize in $\ell^p$ norms, $p\in[1,\infty]$, as the product of the heat kernel, times a $p$-mass function, dependent on the initial condition and $p$. The $p$-mass function is described in terms of boundary averages associated with Busemann functions for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground spherical function. For comparison, the case of the integers shows that a single constant mass determines the asymptotics of solutions to the heat equation for all $p$, emphasizing the influence of the graph geometry on heat diffusion. |
| title | Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees |
| topic | Analysis of PDEs Functional Analysis 35K08, 35B40, 43A85, 43A90, 22E35 |
| url | https://arxiv.org/abs/2603.11232 |