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

We consider the path space of a manifold with measure induced by stochastic flow an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians sub-Riemannian structure chosen complement. introduce concept gradient for cylindrical functionals on in such way operators are closable L2. With this place, we show bound horizontal Ricci curvature equivalent to several inequalities functions space, inequality, log-Sobolev inequality and Poincaré inequality. As consequence, also obtain spectral gap Ornstein–Uhlenbeck operator.