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Functional inequalities on path space of sub-Riemannian manifolds and applications Cheng, Li Juan ; ; Thalmaier, Anton in Nonlinear Analysis: Theory, Methods and Applications (2021), 210(112387), 1-30 We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians ... [more ▼] We consider the path space of a manifold with a measure induced by a stochastic flow with an infinitesimal generator that is hypoelliptic, but not elliptic. These generators can be seen as sub-Laplacians of a sub-Riemannian structure with a chosen complement. We introduce a concept of gradient for cylindrical functionals on path space in such a way that the gradient operators are closable in L^2. With this structure in place, we show that a bound on horizontal Ricci curvature is equivalent to several inequalities for functions on path space, such as a gradient inequality, log-Sobolev inequality and Poincaré inequality. As a consequence, we also obtain a bound for the spectral gap of the Ornstein-Uhlenbeck operator. [less ▲] Detailed reference viewed: 164 (37 UL)Radial processes for sub-Riemannian Brownian motions and applications ; ; et al in Electronic Journal of Probability (2020), 25(paper no. 97), 1-17 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: 195 (22 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: 296 (54 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: 289 (54 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: 145 (7 UL) |
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