Central Gaussian semigroups of measures with continuous density (Q5949999)

The authors investigate continuous symmetric Gaussian convolution semigroups \((\mu_t)_{t\geq 0}\) with \(\mu_0= \delta_e\) on second countable locally compact connected groups \(G\). In particular, groups of the form \(G=\mathbb{R}^n \times K\) are considered, where \(K\) is a compact, connected, locally connected group, and where the case of infinite-dimensional compact Lie groups \(K\) is of special interest. Among other results, the authors give necessary and sufficient conditions on the structure of \(K\) for the existence of symmetric central Gaussian semigroups with continuous densities \(f_t\) for \(t>0\) with respect to the left Haar measure on \(G\). In this case, limit relations like \(\lim_{t\to 0}t^\lambda \cdot\ln f_t(e)=0\) for \(\lambda >0\) and \(\lim_{t\to 0} \sup_{x\in K}f_t(x)=0\) are investigated by splitting the problem on \(G\) into a semisimple part on the commutator subgroup \(G'\) and an abelian part on \(G/G'\). This splitting leads to connections between these limit relations, harmonic sheafs, and quasi-distances on \(G\). For instance, it is shown that the continuity of \(d\) is a sufficient (but not necessary) condition for \(\lim_{t\to 0} t\cdot\ln f_t(e)=0\).