Talagrand-type transport inequalities for path spaces over Carnot groups

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.

💡 Research Summary

This paper establishes Talagrand‑type T₂ transportation inequalities for the law of Brownian motion on step‑2 Carnot groups, with a particular focus on the lifted (enhanced) Brownian motion that lives in the free step‑2 nilpotent group—i.e., the rough path lift. The authors develop several complementary approaches and uncover phenomena that are absent in the Euclidean setting.

First, they introduce a natural cost functional C_H on the path space Ω_G = C₀(