Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group

Abstract

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

Ergodicity of Markov Semigroups with Hörmander Type Generators in Infinite Dimensions

“…After an initial development of a strategy for proving the log-Sobolev inequality for infinite dimensional Hörmander type generators L symmetric in L 2 (µ) defined with a suitable nonproduct measure µ ( [22], [18], [20], [19]), one can envisage an extension of the established strategy (see e.g. [25]) for proving strong pointwise ergodicity for the corresponding Markov semigroups P t ≡ e tL , (or in case of the compact spaces even in the uniform norm as in [14] and references therein).…”

Functional Inequalities and Hamilton–Jacobi Equations in Geodesic Spaces

“…The final section is for remarks and further questions. We also indicate here an application of our results by using a recent result of Inglis and Papageorgiou [12] on the logarithmic Sobolev inequality in the sub-Riemannian setting of the Heisenberg group.…”

Linear and Nonlinear Dissipative Dynamics

“…For symmetric semigroups, after a recent progress in proving the log-Sobolev inequality for infinite dimensional Hörmander type generators L symmetric in L 2 (µ) defined with a suitable nonproduct measure µ ( [32], [25], [28], [26], [27], [43]), one can expect an extension of the established strategy ( [51]) for proving strong pointwise ergodicity for the corresponding Markov semigroups P t ≡ e tL , (respectively in the uniform norm in case of the compact spaces as in [24] and refs therein). One could obtain more results in this direction, including configuration spaces given by infinite products of general noncompact nilpotent Lie groups other than Heisenberg type groups, by conquering a (finite dimensional) problem of subLaplacian bounds (of the corresponding control distance) which for a moment remains still very hard.…”