Uniqueness of embedding into a Gaussian semigroup on a nilpotent Lie group (Q1319597)

It is shown that a Gaussian probability measure has a unique Gaussian embedding convolution semigroup on a simply connected nilpotent Lie group. In other words, if \((\mu_ t)_{t\geq 0}\) and \((\nu_ t)_{t\geq 0}\) are Gaussian semigroups with \(\mu_ 1= \nu_ 1\), then \(\mu_ t= \nu_ t\) for all \(t\geq 0\).