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Talagrand-type transport inequalities for path spaces over Carnot groups

Peter K. Friz, Helena Kremp, Vaios Laschos, Matthias Liero, Benjamin A. Robinson

Abstract

We consider Talagrand-type transportation inequalities for the law of Brownian motion on Carnot groups. An important example is the lift of standard Brownian motion to the Brownian rough path. We present a direct proof on enhanced path space, which also yields equality when restricting to adapted couplings in the transport problem. Moreover, we prove a Talagrand inequality for the heat kernel measure on Carnot groups and deduce the inequality for the law of Brownian motion on Carnot groups via a bottom-up argument. Our study of this enhanced Wiener measure contributes to a longstanding programme to extend key properties of Wiener measure to the non-commutative setting of the enhanced Wiener measure, which is of central importance in Lyons' rough path theory. With a non-commutative sub-Riemannian state space, we observe phenomena that differ from the Euclidean case. In particular, while a top-down projection argument recovers Talagrand's inequality on Euclidean space from the corresponding inequality on the path space, such a projection argument breaks down in the Carnot group setting. We further study a Riemannian approximation of the Heisenberg group, in which case the failure of the top-down projection can be partially overcome. Finally, we show that the cost function used in the Talagrand inequality is a natural choice, in that it arises as a limit of discretised costs in the sense of $Γ$-convergence.

Talagrand-type transport inequalities for path spaces over Carnot groups

Abstract

We consider Talagrand-type transportation inequalities for the law of Brownian motion on Carnot groups. An important example is the lift of standard Brownian motion to the Brownian rough path. We present a direct proof on enhanced path space, which also yields equality when restricting to adapted couplings in the transport problem. Moreover, we prove a Talagrand inequality for the heat kernel measure on Carnot groups and deduce the inequality for the law of Brownian motion on Carnot groups via a bottom-up argument. Our study of this enhanced Wiener measure contributes to a longstanding programme to extend key properties of Wiener measure to the non-commutative setting of the enhanced Wiener measure, which is of central importance in Lyons' rough path theory. With a non-commutative sub-Riemannian state space, we observe phenomena that differ from the Euclidean case. In particular, while a top-down projection argument recovers Talagrand's inequality on Euclidean space from the corresponding inequality on the path space, such a projection argument breaks down in the Carnot group setting. We further study a Riemannian approximation of the Heisenberg group, in which case the failure of the top-down projection can be partially overcome. Finally, we show that the cost function used in the Talagrand inequality is a natural choice, in that it arises as a limit of discretised costs in the sense of -convergence.
Paper Structure (17 sections, 42 theorems, 177 equations)

This paper contains 17 sections, 42 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Preliminaries on Talagrand inequalities
  4. Step-2 Carnot groups
  5. Discussion on the choice of the cost function C_H
  6. Talagrand for Brownian motion on Carnot groups -- Direct approach via Föllmer drift
  7. Talagrand for Gaussian rough paths -- Direct approach via contraction
  8. Talagrand for the heat kernel measure on Carnot groups
  9. From log-Sobolev to Talagrand
  10. From heat semigroup estimates to Talagrand
  11. Talagrand on Heisenberg-type groups
  12. Talagrand for Brownian motion on Carnot groups -- Bottom-up approach
  13. From Talagrand on Carnot groups to Talagrand on path space
  14. Failure of top-down projection and blow-up of cost functions
  15. Riemannian approximation of the Heisenberg group
  16. ...and 2 more sections

Key Result

Theorem 1.1

The measure $\boldsymbol{\mu}$ on $\boldsymbol{\Omega}_\mathbb{G}$ satisfies the $\mathcal{T}_2$ inequality $\boldsymbol{\mu} \in \mathcal{T}_2(\boldsymbol{\Omega}_\mathbb{G}, C_\mathcal{H}, 1)$.

Theorems & Definitions (97)

  • Theorem 1.1: Direct approach, cf. Theorem \ref{['thm:t2-top-foellmer']}, and extensions in \ref{['sec:top-riedel']}
  • Theorem 1.2: Bottom-up, cf. Theorem \ref{['thm:t2-carnot-path']}
  • Theorem 1.3: $\mathcal{T}_2$ on group, cf. Theorem \ref{['thm:t2-h-type']}
  • Theorem 1.4: Adapted couplings, cf. Theorem \ref{['thm:t2-top-foellmer']}
  • Theorem 1.5: Cost approximation, cf. \ref{['cor:gamma-convergence', 'thm:GammaConvTransport']}
  • Theorem 1.6: Top-down -- validity vs. failure
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 87 more