Centered sub-Laplacians and densities in Lie groups of polynomial volume growth

We prove a Harnack inequality on connected Lie groups of polynomial volume growth. We use this inequality to study the large time behavior of the heat kernels associated to centered sub-laplacians. Thus, we obtain Gaussian estimates and estimates of the type Berry-Esseen. We also obtain similar results of the convolution powers of centered densities. (C) Academie des Sciences/Elsevier, Paris.