Oscillating heat kernels on ultrametric spaces

Abstract

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 $\mathfrak{p}(t)=t^{-1/\alpha }\mathcal{A}({\log }_{p}t)$, 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 $t\to \infty$.