We prove a parabolic Harnack inequality for a centered sub-Laplacian L on a con-
nected Lie group G of polynomial volume growth by using ideas from Homogeni-
sation theory and by adapting the method of Kryiov and Safonov. We use this
inequality to obtain a Taylor formula for the heat functions and thus we also ob-
tain Harnack inequalities for their space and time derivatives. We characterise the
harmonic functions which grow polynomially. We obtain Gaussian estimates for the
heat kernel and estimates similar to the classical Berry-Esseen estimate. Finally, we
study the associated Riesz transform operators. If L is not centered, then we can
conjugate L by a convenient multiplicative function and obtain another centered
sub-Laplacian Lc- Thus our results also extend to non-centered sub-Laplacians .
2000 Mathematics Subject Classification. 22E15, 22E25, 22E30, 43A80.
Key words and phrases. Lie group, volume growth, sub-Laplacian, drift, homogenised oper-
ator, harmonic function, Harnack inequality, heat kernel, Riesz transform.