Positive curvature property for some hypoelliptic heat kernels |
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Authors: | Bin Qian |
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Institution: | aSchool of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, PR China;bInstitut de Mathématiques de Toulouse, Université de Toulouse, CNRS 5219, France |
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Abstract: | In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. |
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Keywords: | MSC: 58J35 43A80 |
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