Radial processes for sub-Riemannian Brownian motions and applications.

*(English)*Zbl 1451.53046The authors study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô’s formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. They 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, can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, they also 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.

Reviewer: Peibiao Zhao (Nanjing)

##### MSC:

53C17 | Sub-Riemannian geometry |

35H20 | Subelliptic equations |

58J65 | Diffusion processes and stochastic analysis on manifolds |

##### Keywords:

sub-Riemannian Brownian motion; radial process; stochastic completeness; sub-Laplacian comparison theorem; Sasakian manifold; \(H\)-type group; Riemannian foliation
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\textit{F. Baudoin} et al., Electron. J. Probab. 25, Paper No. 97, 17 p. (2020; Zbl 1451.53046)

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