Positive curvature property for sub-Laplace on nilpotent Lie group of rank two

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 $\frac{n(n-1)}{2}$ L'evy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$. 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 \ref{radial fun}) satisfies a positive Ricci curvature condition (more precisely a $CD(0,\mathrm{\infty })$ inequality, see Theorem \ref{positive}, whereas the operator itself does not satisfy any $CD(r,\mathrm{\infty })$ 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 \cite{Qian2}.