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See detailFunctional inequalities on path space of sub-Riemannian manifolds and applications
Cheng, Li Juan UL; Grong, Erlend; Thalmaier, Anton UL

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 ▲]

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See detailStochastic completeness and gradient representations for sub-Riemannian manifolds
Grong, Erlend; Thalmaier, Anton UL

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 ▲]

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See detailSub-Laplacian comparison theorems on totally geodesic Riemannian foliations
Baudoin, Fabrice; Grong, Erlend; Kuwada, Kazumasa 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 ▲]

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See detailICAMI 2017: International Conference on Applied Mathematics and Informatics: Forum on Analysis, Geometry, and Mathematical Physics
Schlichenmaier, Martin UL; Molina, Goday; Grong, Erlend et al

in Analysis and Mathematical Physics (2018), 8

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