2011/12/15 by Bin Qian, Qian, Bin
Mathematics · #58J35 #60J60 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Probability (math.PR) #math.AP #math.DG #math.PR #msc:58J35 #msc:60J60
paper · pdf · doi:10.48550/arxiv.1112.3402
arxiv created 2011/12/15 · openalex publication_date 2011/12/15 · arxiv updated 2011/12/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by n Brownian motions and their \fracn(n-1)2 Lévy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group ℍ. In order to study contraction properties of the heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motion model, the restriction of the sub-Laplace acting on radial functions (see Definition \refradial fun) satisfies a positive Ricci curvature condition (more precisely a CD(0, ∞) inequality, see Theorem \refpositive, 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. It can be seen a generalization of the paper \citeQian2.