Talagrand-type transport inequalities for path spaces over Carnot groups

TL;DR

This paper establishes Talagrand-type transportation inequalities for Brownian motion on Carnot groups, extending key properties of Wiener measure to non-commutative sub-Riemannian spaces, with implications for rough path theory.

Contribution

It provides a direct proof of Talagrand inequalities on path spaces over Carnot groups and explores the limitations of projection methods in non-commutative settings.

Findings

Proved Talagrand inequalities for heat kernel measures on Carnot groups.

Identified breakdown of projection arguments in non-commutative spaces.

Showed the cost function as a limit of discretised costs via $ extGamma$-convergence.

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 $\Gamma$-convergence.