Functional inequalities on path space of sub-Riemannian manifolds and applications

TL;DR

This paper develops functional inequalities on the path space of sub-Riemannian manifolds by establishing derivative and integration by parts formulas, linking curvature bounds to inequalities like log-Sobolev and Poincaré.

Contribution

It introduces new derivative and integration by parts formulas on sub-Riemannian path space and connects curvature bounds to functional inequalities, extending Riemannian results.

Findings

Derivative and integration by parts formulas established

Curvature bounds imply functional inequalities

Extension of Riemannian geometric inequalities to sub-Riemannian setting

Abstract

For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincar\'e inequalities.