The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by $$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$ where $\unicode[STIX]{x1D6EF}$ is the space of choice functions on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$. There are two ultrametrics, $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$, on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ which make the infinite path space $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of the Laplace–Beltrami operator $\unicode[STIX]{x1D6E5}_{s}$ associated to $Q_{s}$, and the latter $d_{w_{\unicode[STIX]{x1D6FF}}}$ is associated to a weight function $w_{\unicode[STIX]{x1D6FF}}$ on ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$. We show that the Perron–Frobenius measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ has the volume-doubling property with respect to both $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ and we study the asymptotic behavior of the heat kernel associated to $Q_{s}$. Moreover, we show that the Dirichlet form $Q_{s}$ coincides with a Dirichlet form ${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$ which is associated to a jump kernel $J_{s}$ and the measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, and we investigate the asymptotic behavior and moments of displacements of the process.

中文翻译：

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与高阶图相关的狄利克雷形式和超量康托集

本文的目的是研究与超度量康托集相关的狄利克雷形式的热核和跳跃核$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$那是静止的无限路径空间$k$-布拉特利图${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$， 在哪里$\unicode[STIX]{x1D6EC}$是一个有限强连接$k$-图形。我们感兴趣的狄利克雷形式是由偶谱三元组引起的$(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\ unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$并且由$$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{- s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX ]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$在哪里$\unicode[STIX]{x1D6EF}$是选择函数的空间$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX ]{x1D6EC}}$. 有两种超度量，$d^{(s)}$和$d_{w_{\unicode[STIX]{x1D6FF}}}$， 在$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$这使得无限路径空间$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$超度量康托集。前者$d^{(s)}$与 Laplace–Beltrami 算子的特征值相关联$\unicode[STIX]{x1D6E5}_{s}$关联到$Q_{s}$, 后者$d_{w_{\unicode[STIX]{x1D6FF}}}$与权重函数相关联$w_{\unicode[STIX]{x1D6FF}}$在${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$， 在哪里$\unicode[STIX]{x1D6FF}\in (0,1)$. 我们证明了 Perron-Frobenius 测度$\unicode[STIX]{x1D707}$在$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$两者都具有体积倍增特性$d^{(s)}$和$d_{w_{\unicode[STIX]{x1D6FF}}}$我们研究了与相关的热核的渐近行为$Q_{s}$. 此外，我们证明了狄利克雷形式$Q_{s}$符合狄利克雷形式${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$与跳转内核相关联$J_{s}$和措施$\unicode[STIX]{x1D707}$在$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, 我们研究了过程的渐近行为和位移矩。