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Dimension-independent functional inequalities by tensorization and projection arguments

Fabrice Baudoin, Maria Gordina, Rohan Sarkar

TL;DR

The paper develops a framework for dimension-independent functional inequalities for Markov semigroups via tensorization and projection on metric spaces. By leveraging transportation-type inequalities and duality results, it proves that gradient bounds, reverse Poincaré, and reverse logarithmic Sobolev inequalities persist with constants independent of dimension for product spaces and under submersion, extending to sub-Riemannian manifolds and a broad class of hypoelliptic diffusions. The results yield Li–Yau type gradient estimates and parabolic Harnack inequalities in tensorized/submersion settings and provide explicit, sharp constants in canonical examples such as Kolmogorov diffusions, kinetic Fokker–Planck equations, and Lie groups with transverse symmetry, including SO(3), SO(4), and Heisenberg-type spaces. These dimension-free bounds have potential implications for high- and infinite-dimensional diffusion problems and broad applicability across geometric and probabilistic contexts.

Abstract

We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation-type inequalities obtained by F. Baudoin and N. Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension-independent reverse Poincaré, reverse logarithmic Sobolev, and gradient bounds for Lie groups with a transverse symmetry and for non-isotropic Heisenberg groups.

Dimension-independent functional inequalities by tensorization and projection arguments

TL;DR

The paper develops a framework for dimension-independent functional inequalities for Markov semigroups via tensorization and projection on metric spaces. By leveraging transportation-type inequalities and duality results, it proves that gradient bounds, reverse Poincaré, and reverse logarithmic Sobolev inequalities persist with constants independent of dimension for product spaces and under submersion, extending to sub-Riemannian manifolds and a broad class of hypoelliptic diffusions. The results yield Li–Yau type gradient estimates and parabolic Harnack inequalities in tensorized/submersion settings and provide explicit, sharp constants in canonical examples such as Kolmogorov diffusions, kinetic Fokker–Planck equations, and Lie groups with transverse symmetry, including SO(3), SO(4), and Heisenberg-type spaces. These dimension-free bounds have potential implications for high- and infinite-dimensional diffusion problems and broad applicability across geometric and probabilistic contexts.

Abstract

We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation-type inequalities obtained by F. Baudoin and N. Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension-independent reverse Poincaré, reverse logarithmic Sobolev, and gradient bounds for Lie groups with a transverse symmetry and for non-isotropic Heisenberg groups.
Paper Structure (17 sections, 19 theorems, 125 equations)

This paper contains 17 sections, 19 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Tensorization of functional inequalities on metric measure spaces
  3. Transportation inequalities and tensorization
  4. Applications to sub-Riemannian manifolds
  5. Sub-Riemannian manifolds obtained by tensorization and projection
  6. Carnot-Carathéodory metric on manifolds obtained by tensorization and submersions
  7. Proof of Theorem \ref{['thm:tensorization']}
  8. Proof of Theorem \ref{['thm:projection']}
  9. Examples
  10. Kolmogorov diffusion on Euclidean spaces
  11. Kinetic Fokker-Planck equations
  12. Lie groups with transverse symmetries
  13. Step $2$ homogeneous Carnot group
  14. Hypoelliptic heat equation on $\operatorname{SO}(3)$
  15. Hypoelliptic heat equation on $\operatorname{SO}(4)$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $P_i$ be a Markov kernel on $(X_i, d_i), i=1, ..., n$ that satisfies eq:GB (resp. eq:RP, eq:RLS) with a constant $C_i$. Then the Markov kernel $P:=\otimes_{i=1}^n P_i$ satisfies eq:GB (resp. eq:RP, eq:RLS) with constant $C=\max\left\{C_i: 1 \leqslant i\leqslant n \right\}$.

Theorems & Definitions (46)

  • Theorem 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Definition 3.1
  • Remark 3.2
  • Example 3.3: Riemannian submersions
  • Example 3.4
  • Example 3.5: Sub-Riemannian isometries
  • ...and 36 more