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

Authors
  • Cheng, Li Juan
  • Grong, Erlend
  • Thalmaier, Anton
Publication Date
Sep 01, 2021
Source
ORBilu
Keywords
Language
English
License
Green
External links

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.

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