# Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group

## Details

State: Public
Version: Final published version
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.
Serval ID
serval:BIB_E9F5CC83740E
Type
Article: article from journal or magazin.
Collection
Publications
Institution
Title
Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group
Journal
Journal of Theoretical Probability
ISSN
0894-9840
Publication state
Published
Issued date
2008
Peer-reviewed
Oui
Volume
21
Number
4
Pages
791-801
Language
english
Abstract
Let {mu((i))(t)}(t >= 0) (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that mu((1))(1)=mu((2))(1). Assume furthermore that one of the following two conditions holds:
(i) The c.c.s. {mu((1))(t)}(t >= 0) is a Gaussian semigroup (in the sense that its generating distribution just consists of a primitive distribution and a second-order differential operator)
(ii) The c.c.s. {mu((i))(t)}(t >= 0) (i=1,2) are both Poisson semigroups, and the jump measure of {mu((1))(t)}(t >= 0) is determinate (i.e., it possesses all absolute moments, and there is no other nonnegative bounded measure with the same moments).
Then mu((1))(t) = mu((2))(t) for all t >= 0. As a complement, we show how our approach can be directly used to give an independent proof of Pap's result on the uniqueness of the embedding Gaussian semigroup on simply connected nilpotent Lie groups. In this sense, our proof for the uniqueness of the embedding semigroup among all c.c.s. of a Gaussian measure can be formulated self-contained.
Keywords
Continuous convolution semigroups · Simply connected nilpotent Lie groups · Gaussian semigroups · Poisson semigroups
Web of science
Open Access
Yes
Create date
08/02/2010 17:03
Last modification date
14/02/2022 8:57
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