Symmetric stable semigroups on the infinite dimensional torus
Let \((\mu_t)_{ t>0}\) be the Brownian semigroup on the torus \(T\). For each sequence \({\mathcal A}= \{a_k \}_1^\infty\) of strictly positive numbers, and for each \(t>0\), let \(\mu_t^{\mathcal A}= \otimes_{k=1}^\infty \mu_{ta_k}\) be the product measure on the infinite dimensional torus \(T^\infty\). For each \(\alpha\in (0,1)\), let \((\mu_t^{{\mathcal A}, \alpha} )_{t> 0}\) be the convolution semigroup on \(T^\infty\) which is subordinated to \((\mu_t^{\mathcal A} )_{t> 0}\) by means of the one-sided stable semigroup on \(\mathbb{R}\) of index \(\alpha\). This semigroup, called the symmetric stable semigroup, generates a satisfactory potential theory provided the following two properties are satisfied: (i) For each \(t>0\), the measure \(\mu_t^{{\mathcal A}, \alpha}\) is absolutely continuous with respect to the Haar measure, and (ii) for each \(t>0\) the density \(\mu_t^{{\mathcal A},\alpha} (x)\) is a continuous function. These two properties are the main object of the paper under review. Continuity of the density \(\mu_t^{{\mathcal A},\alpha} (x)\) depends on the asymptotics of the counting function \(N^{\mathcal A} (\lambda)= \sum_{a_k\leq \lambda} 1\) as \(\lambda\to \infty\). One of the author's results is that \(\mu_t^{{\mathcal A},\alpha} (x)\) is continuous for every \(t>0\) if and only if \(N^{\mathcal A} (\lambda)= o(\lambda^{\alpha/ (1- \alpha)})\) as \(\lambda\to \infty\). Several precise sufficient conditions for continuity are also given. One of the crucial points of the author's investigation is the identity \[ \log \mu_t^{\mathcal A} (0)= k* N^{\mathcal A} (1/t) \] where \(*\) denotes the Mellin convolution on \(\mathbb{R}\), and \(k\) is a function not depending on the semigroup. This identity gives precise information on the decay of the function \(t\mapsto \mu_t^{\mathcal A}\) as \(t\to 0\).
scientific article; zbMATH DE number 783956