Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs

The aim of this paper is to study the heat kernel and jump kernel of the Dirichlet form associated to ultrametric Cantor sets $\mathrm{\partial }{\text{BB}}_{\mathrm{\Lambda }}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\text{BB}}_{\mathrm{\Lambda }}$, where $\mathrm{\Lambda }$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\mathrm{Lip}}({\text{PB}}_{\mathrm{\Lambda }}),{\pi }_{\varphi },\mathcal{H},D,\mathrm{\Gamma })$ and is given by $$Q_s(f,g)=\frac{1}{2}{\int }_{\mathrm{\Xi }}\mathrm{Tr}(\mathrm{\mid }D{\mathrm{\mid }}^{-s}[D,{\pi }_{\varphi }(f){]}^{\ast }[D,{\pi }_{\varphi }(g)])d\nu (\varphi ),$$ where $\mathrm{\Xi }$ is the space of choice functions on $\partial \BB_\Lambda \times \partial \BB_\Lambda$\mathrm{.}Therearetwoultrametrics,$d^{(s)}$and$d_{w_\delta}$,on$\partial \BB_\Lambda$whichmaketheinfinitepathspace$\PB_\Lambda$anultrametricCantorset\mathrm{.}Theformer$d^{(s)}$ is associated to the eigenvalues of Laplace-Beltrami operator ${\mathrm{\Delta }}_s$ associated to $Q_s$, and the latter $d_{w_{\delta }}$ is associated to a weight function $w_{\delta }$ on ${\text{BB}}_{\mathrm{\Lambda }}$, where $\delta \in (0,1)$. We show that the Perron-Frobenius measure $\mu$ on $\mathrm{\partial }{\text{BB}}_{\mathrm{\Lambda }}$ has the volume doubling property with respect to both $d^{(s)}$ and $d_{w_{\delta }}$ and we study the asymptotic behaviors of the heat kernel associated to $Q_s$. Moreover, we show that the Dirichlet form $Q_s$ coincides with a Dirichlet form ${\mathcal{Q}}_{J_s,\mu }$ which is associated to a jump kernel $J_s$ and the measure $\mu$ on $\partial \BB_\Lambda$, and we investigate the asymptotic behavior and moments of displacements of the process.