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

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It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.

State: Public

Version: Final published version

License: Not specified

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.

(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