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Curvature and Other Local Inequalities in Markov Semigroups

Devraj Duggal, Andreas Malliaris, James Melbourne, Cyril Roberto

Abstract

Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local functional inequalities. We will show that not only is the earlier approach for strictly convex potentials extendable to Markov semigroups and simplified through use of the $Γ$-calculus, providing a consolidating machinery for obtaining functional inequalities new and old in this general setting, but that a converse also holds. Local inequalities obtained are proven equivalent to Bakry-Emery curvature. Moreover we will develop this technique for metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semi-group operators on $\mathbb{R}^n$ in the absence of positive curvature, based on a local eigenvalue criteria.

Curvature and Other Local Inequalities in Markov Semigroups

Abstract

Inspired by the approach of Ivanisvili and Volberg towards functional inequalities for probability measures with strictly convex potentials, we investigate the relationship between curvature bounds in the sense of Bakry-Emery and local functional inequalities. We will show that not only is the earlier approach for strictly convex potentials extendable to Markov semigroups and simplified through use of the -calculus, providing a consolidating machinery for obtaining functional inequalities new and old in this general setting, but that a converse also holds. Local inequalities obtained are proven equivalent to Bakry-Emery curvature. Moreover we will develop this technique for metric measure spaces satisfying the RCD condition, providing a unified approach to functional and isoperimetric inequalities in non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are interested in commutation properties for semi-group operators on in the absence of positive curvature, based on a local eigenvalue criteria.
Paper Structure (17 sections, 23 theorems, 202 equations)

This paper contains 17 sections, 23 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. The underlying architecture
  3. Markov triple
  4. Bakry-Emery condition
  5. $RCD(\rho,\infty)$ spaces
  6. Curvature as a cornerstone of semi-group interpolation
  7. Local inequalities
  8. Integrated condition
  9. Local inequalities in $RCD(\rho,\infty)$ spaces
  10. Dimensional considerations and higher order operators
  11. Generalised Local Inequalities
  12. Ledoux-Houdré-Kagan's inequality
  13. Interpolation under $BE(\rho,n)$
  14. A second look at $BE(0,\infty)$
  15. A study of the local eigenvalue condition via examples
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(E,\mu,\Gamma)$ be a Markov triple. Let $\rho \in \mathbb{R}$ and $M \colon I \times \mathbb{R}^+$ be of class $\mathcal{C}^2$, for an interval $I \subseteq \mathbb{R}$, with $M_y \geq 0$ on its domain and not identically $0$. For $t , \alpha \geq 0$, define $g_\alpha(t) \coloneqq \frac{1-e^{-2 is positive semi-definite. Then the following are equivalent: $(i)$ For any $f \in \mathcal{A}$ it

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Sturm-Lott-Villani
  • Definition 2.2: $RCD$-spaces AGS
  • Theorem 2.3: Ambrosio-Gigli-Savare
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Theorem 3.3
  • ...and 37 more