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 L2. 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.
Highlights
Stochastic analysis on the path space over a complete Riemannian manifold has been well developed ever since B
A key point of the study is to first establish an integration by parts formula for the associated gradient operator induced by the quasi-invariant flows, prove functional inequalities for the corresponding Dirichlet form
Nonlinear Analysis 210 (2021) 112387. We develop this formalism in the framework of hypoelliptic operators and diffusions in sub-Riemannian geometry
Summary
Stochastic analysis on the path space over a complete Riemannian manifold has been well developed ever since B. In spite of the fact that this adjoint will not be compatible with the sub-Riemannian structure, we show that the gradient and the damped gradient are related by the Ricci operator Having set up this formalism, we extend the approach of Naber to the sub-Riemannian case in our main result in Theorem 4.1. We show that if one uses the canonical connection ∇ corresponding to a metric preserving complement V , the sub-Riemannian path space has geometry “similar to M/V ” This latter concept is well defined in the case when V is an integrable submanifold corresponding to a regular foliation Φ in which M/Φ has an induced Riemannian structure, but our formalism is valid for non-integrable choices of complements as well. To explain the reason behind this assumption and for later references, we include some formulas related to a general choice of connection and complement in the Appendix
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