Convolution roots and embeddings of probability measures on Lie groups

On rationally embeddable measures on Lie groups

“…This is now known as the "embedding problem" and has been the subject of much study in literature. For references on the work, the reader is referred to Dani and McCrudden (2007) where the result has been established for a large class of Lie groups, and also the recent articles McCrudden (2006) and Dani (2009); the general question remains open however.…”

Convolution roots and embeddings of probability measures on locally compact groups

“…The above mentioned conjecture for connected Lie groups was established in [5] for a large class of Lie groups, called class C groups, including all (connected) linear Lie groups, all simply connected Lie groups and all semisimple Lie groups. A more transparent approach to the problem was introduced in [6]. The focus of the proof in [6] nevertheless remained on algebraic groups, from which the results were deduced by extension to the class C groups.…”

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.…”