Central Gaussian Semigroups of Measures with Continuous Density

Abstract

This paper investigates the existence and properties of symmetric central Gaussian semigroups (μt)t>0 which are absolutely continuous and have a continuous density x↦μt(x), t>0, with respect to Haar measure on groups of the form Rn×K where K is compact connected locally connected and has a countable basis for its topology. We prove that there always exists a wealth of such Gaussian semigroups on any such group. For instance, if ψ is any positive function increasing to infinity, there exists a symmetric central Gaussian semigroup having a continuous density such that logμt(e)⩽log(1+1/t)ψ(1/t) as t tends to zero. Among other results of this type we give a necessary and sufficient condition on the structure of K for the existence of symmetric central Gaussian semigroups having a continuous density and such that tλlogμt(e) is bounded above and below by positive constants for t∈(0, 1) and some fixed λ>0. This condition is independent of λ. These results are proved by splitting any Gaussian semigroup (in a canonical way) into a semisimple part living on the commutator group G′ and an Abelian part living on A=G/G′. For symmetric central Gaussian semigroups, we show that many properties hold for (μt)t>0 if and only if they hold for both the semisimple and the Abelian parts. This splitting principle is one of the main new tools developed in this paper. It leads to a much better understanding of central Gaussian semigroups and related objects and lets us answer a number of open questions. For instance, to any Gaussian convolution semigroup are associated a harmonic sheaf H and a quasi-distance d on G. For symmetric central Gaussian semigroups on G=Rn×K, we show that H is a Brelot sheaf if and only if limt→0tlogμt(e)=0. A sufficient (but not necessary) condition is that the distance d is continuous. Together with a celebrated result of Bony, our results show that the compact group K is a Lie group if and only if any bi-invariant elliptic Bauer harmonic sheaf on G is a Brelot sheaf.