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Radial processes for sub-Riemannian Brownian motions and applications ; ; et al E-print/Working paper (2020) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 21 (4 UL)Functional inequalities on path space of sub-Riemannian manifolds and applications Cheng, Li Juan ; ; Thalmaier, Anton E-print/Working paper (2019) For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a well-defined gradient operator. By using ... [more ▼] For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a well-defined gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequalities on path space analogous to what has been established in Riemannian geometry by Aaron Naber, such as gradient inequalities, log-Sobolev and Poincaré inequalities. [less ▲] Detailed reference viewed: 29 (4 UL)Stochastic completeness and gradient representations for sub-Riemannian manifolds ; Thalmaier, Anton in Potential Analysis (2019), 51(2), 219-254 Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a ... [more ▼] Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup P_t, we give two different stochastic representations of dP_t f for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of L/2, while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry. [less ▲] Detailed reference viewed: 255 (53 UL)Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations ; ; et al in Calculus of Variations and Partial Differential Equations (2019), 58:130(4), 1-38 We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical ... [more ▼] We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations. [less ▲] Detailed reference viewed: 234 (52 UL)ICAMI 2017: International Conference on Applied Mathematics and Informatics: Forum on Analysis, Geometry, and Mathematical Physics Schlichenmaier, Martin ; ; et al in Analysis and Mathematical Physics (2018), 8 Detailed reference viewed: 104 (7 UL) |
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