Local limit theorem of Brownian motion on metric trees
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914708015022080 |
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| author | Hong, Soonki |
| author_facet | Hong, Soonki |
| contents | Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group $\G$ acts isometrically and geometrically on the tree $\mathcal{T}$.
In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function $C$ on $\mathcal{T}^2$ such that the heat kernel $p(t,x,y)$ of $\mathcal{T}$ satisfies
$$\lim_{t\rightarrow \infty}t^{3/2}e^{λ_0t}p(t,x,y)=C(x,y)$$
for any $x,y\in \mathcal{T}$. Here, $λ_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local limit theorem of Brownian motion on metric trees Hong, Soonki Dynamical Systems Probability 37H05, 31C25, 37D35 Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has infinitely many points. Suppose that a non-amenable group $\G$ acts isometrically and geometrically on the tree $\mathcal{T}$. In this paper, we show that if the length spectrum is Diophantine, then there exists a continuous function $C$ on $\mathcal{T}^2$ such that the heat kernel $p(t,x,y)$ of $\mathcal{T}$ satisfies $$\lim_{t\rightarrow \infty}t^{3/2}e^{λ_0t}p(t,x,y)=C(x,y)$$ for any $x,y\in \mathcal{T}$. Here, $λ_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$. |
| title | Local limit theorem of Brownian motion on metric trees |
| topic | Dynamical Systems Probability 37H05, 31C25, 37D35 |
| url | https://arxiv.org/abs/2403.05089 |