Stochastic completeness and gradient representations for Sub-Riemannian manifolds. (English) Zbl 1481.60156
Summary: 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 \(\frac{1}{2} L\), 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.
MSC:
| 60J60 | Diffusion processes |
| 35P99 | Spectral theory and eigenvalue problems for partial differential equations |
| 53C17 | Sub-Riemannian geometry |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
Keywords:
diffusion process; stochastic completeness; hypoelliptic operators; gradient bound; sub-Riemannian geometryReferences:
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