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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: 224 (52 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: 198 (49 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: 80 (6 UL) |
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