Talagrand-type transport inequalities for path spaces over Carnot groups
Authors
- Friz, Peter K.
ORCID: 0000-0003-2571-8388 - Kremp, Helena
- Laschos, Vaios
ORCID: 0000-0001-8721-5335 - Liero, Matthias
ORCID: 0000-0002-0963-2915 - Robinson, Benjamin A.
2020 Mathematics Subject Classification
- 39B62 49Q22 53C17 60E15 60L20
Keywords
- Carnot groups, log-Sobolev inequality, optimal transport, rough paths, Talagrand inequality, transportation inequalities
DOI
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.
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