Logarithmic Sobolev inequalities for infinite dimensional H\"ormander type generators on the Heisenberg group

The Heisenberg group is one of the simplest sub-Riemannian settings in which we can define non-elliptic H"ormander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities.