Radial processes for sub-Riemannian Brownian motions and applications
by Fabrice Baudoin, Erlend Grong, Kazumasa Kuwada, Rob Neel and Anton Thalmaier
Abstract
We 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. 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.
Electron. J. Probab. 25 (2020), paper no. 97, 17 pp.
https://doi.org/10.1214/20-EJP5017
The paper is available here:
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