Ornstein–Uhlenbeck semi-groups on stratified groups

We consider, in the setting of stratified groups G, two analogues of the Ornstein–Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on Lq(pdγ ), whose invariant density p is a heat kernel at time 1 on G. Both generators have the same “carré du champ”. The first semi-group is symmetric on L2(pdγ ), with generator n i=1 X∗i Xi, where (Xi )n i=1 is a basis of the first layer of the Lie algebra of G. The second one is compact on Lq(pdγ ), 1