Abstract
We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Ito’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng’s type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.
Highlights
In the context of Riemannian manifolds the study of the radial part of Brownian motion yields new proofs and sheds new light on several well-known theorems of Riemannian geometry; see for instance the paper [20] and the book [19] for an overview
The first one is the class of sub-Riemannian manifolds whose horizontal distribution is the horizontal distribution of some Riemannian foliation with totally geodesic leaves
One of the main results we obtain for the radial processes is Theorem 3.1, giving its Itô formula
Summary
In the context of Riemannian manifolds the study of the radial part of Brownian motion yields new proofs and sheds new light on several well-known theorems of Riemannian geometry; see for instance the paper [20] and the book [19] for an overview. Let M be a sub-Riemannian manifold associated to a Sasakian foliation, with horizontal distribution of dimension n, and which satisfies the above comparison result with k1 = k2 = 0. Such a representation allows us to give a criterion for non-explosion of the sub-Riemannian Brownian motion, which is more general than previous criteria for stochastic completeness found in [8] and [18].
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