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Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that if $L$ satisfies, with a non negative curvature parameter $\rho_1$, the generalized curvature inequality in \eqref{CD} below, then the Riesz transform is bounded in $L^p (\bM)$ for every $p>1$, that is \[\| \sqrt{\Gamma((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \] where $\Gamma$ is the \textit{carr\'e du champ} associated to $L$. Our results apply in particular to all Sasakian manifolds whose horizontal Tanaka-Webster Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU(2), Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\Gamma_{2}$ techniques that we adapt to those elementary model spaces. The important feature developed here is that although the usual notion of Ricci curvature is meaningless (or more precisely leads to bounds of the form $-\infty$ for the Ricci curvature), we describe a parameter $\rho$ which plays the same role as the lower bound on the Ricci curvature, and from which one deduces the same kind of results as one does in Riemannian geometry, like heat kernel upper bounds, Sobolev inequalities and diameter estimates.

It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy H\"ormander's condition. The main new ingredient of the proof is an extension of Norris' lemma to this situation.

The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized. Finally, we prove an integration by parts formula on the path group space and deduce the existence of a density.