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Positive curvature property for some hypoelliptic heat kernels

Authors
Journal
Bulletin des Sciences Mathématiques
0007-4497
Publisher
Elsevier
Publication Date
Volume
135
Issue
3
Identifiers
DOI: 10.1016/j.bulsci.2010.08.001
Keywords
  • [Formula Omitted]Curvature
  • Heat Kernel
  • Gradient Estimates
  • Sublaplacian
  • Three Brownian Motions Model
Disciplines
  • Mathematics

Abstract

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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