Oscillating heat kernels on ultrametric spaces

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\,$. The operator $L$ acts in $\mathcal{L}^{2}(X,m),$ is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel $\mathfrak{p}(t,x,y)$ with respect to $m$. We consider the case when $X$ has a transitive group of isometries under which the operator $L$ is invariant and study the asymptotic behaviour of the function $t\mapsto \mathfrak{p}(t,x,x)=\mathfrak{p}(t)$. It is completely monotone, but does not vary regularly. When $X=\mathbb{Q}_{p}\,$, the ring of $p$-adic numbers, and $L=\mathcal{D}^{\alpha}$, the operator of \ fractional derivative of order $\alpha,$ we show that $\mathfrak{p}(t)=t^{-1/\alpha}\mathcal{A}$, where $\mathcal{A}(\tau)$ is a continuous non-constant $\alpha$-periodic function. We also study asymptotic behaviour of $\min\mathcal{A}$ and $\max\mathcal{A}$ as the space parameter $p$ tends to $\infty$. When $X=S_{\infty}\,$, the infinite symmetric group, and $L$ is a hierarchical Laplacian with metric structure analogous to $\mathcal{D}^{\alpha},$ we show that, contrary to the previous case, the completely monotone function $\mathfrak{p}(t)$ oscillates between two functions $\psi(t)$ and $\Psi(t)$ such that $\psi(t)/\Psi(t)\to 0$ as $t \to \infty\,$.