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Oscillating heat kernels on ultrametric spaces. (English) Zbl 1409.60114

Summary: 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 \(\psi(t)/\Psi(t)\rightarrow 0\) as \(t \rightarrow \infty\).

MSC:

60J35 Transition functions, generators and resolvents
12H25 \(p\)-adic differential equations
20K25 Direct sums, direct products, etc. for abelian groups
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
60J25 Continuous-time Markov processes on general state spaces
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