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

Let {μ t (i) } t≥0 (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that μ 1 (1) =μ 1 (2) . Assume furthermore that one of the following two conditions holds: (i) The c.c.s. {μ t (1) } 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. {μ t (i) } t≥0 (i=1,2) are both Poisson semigroups, and the jump measure of {μ t (1) } t≥0 is determinate (i.e., it possesses all absolute moments, and there is no other nonnegative bounded measure with the same moments). Then μ t (1) =μ t (2) 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.