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

TL;DR

This paper proves that certain Gibbs measures with quadratic interactions on infinite-dimensional Heisenberg groups satisfy logarithmic Sobolev inequalities, advancing understanding of functional inequalities in sub-Riemannian infinite-dimensional settings.

Contribution

It establishes logarithmic Sobolev inequalities for Gibbs measures on infinite product Heisenberg groups with non-elliptic H"ormander generators, a novel result in this context.

Findings

Gibbs measures satisfy logarithmic Sobolev inequalities

Results extend inequalities to infinite-dimensional sub-Riemannian settings

Advances understanding of functional inequalities for non-elliptic operators

Abstract

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.