On the embeddability of certain infinitely divisible probability measures on Lie groups

Abstract

International audienceWe describe certain sufficient conditions for an infinitely divisible probability measure on a Lie group to be embeddable in a continuous one-parameter semigroup of probability measures. A major class of Lie groups involved in the analysis consists of central extensions of almost algebraic groups by compactly generated abelian groups without vector part. This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain connected Lie groups, including the so called Walnut group. The embeddability is concluded also under certain other conditions. Our methods are based on a detailed study of actions of certain nilpotent groups on special spaces of probability measures and on Fourier analysis along the fibering of the extension

Convolution of Probability Measures on Lie Groups and Homogenous Spaces

“…This shows that µ satisfies (5). Conversely, using the K-right invariance of ρ K , it is easy to show that µ given by (5) is K-right invariant with πµ = ν.…”

“…We note that the embedding may hold on Lie groups that are not of class C, even some locally compact groups, see for example [5] for some of such Lie groups. It may be possible to obtain the embedding on the homogeneous spaces of these groups by the present approach.…”

Spectral properties of a class of random walks on locally finite groups

“…The problem of embedding an infinite divisible probability measure µ, defined on a general locally compact group, into a weakly continuous convolution semigroup (µ t ) t>0 is open. See the papers of S.G. Dani, Y. Guivarc'h, and R. Shah [16], and McCrudden [39]. We provide a solution to this problem when the underlying group G is locally finite and the measure µ is a convex linear combination of idempotent measures.…”