Local limit theorem of Brownian motion on metric trees

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 $$\lim_{t\rightarrow \infty}t^{3/2}e^{\lambda_0t}p(t,x,y)=C(x,y)$$ for any $x,y\in \mathcal{T}$. Here, $\lambda_0$ is the bottom of the spectrum of the Laplacian on $\mathcal{T}$.