Functional inequalities on path space of sub-Riemannian manifolds and applications. (September 2021)

• Record Type: Journal Article
• Authors: Cheng, Li-Juan; Grong, Erlend; Thalmaier, Anton
• Is Part Of: Nonlinear analysis. Volume 210(2021)
• Publication Date: 2021-09
• Subjects: 60J60 -- 58J65 -- 53C17; Bounded Ricci curvature -- Path space -- Functional inequalities -- Sub-Riemannian geometry; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112387
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in L 2 . With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein–Uhlenbeck operator.