In this talk, I will discuss the problem of defining in sub-Riemannian
geometry an hypoelliptic operator generalizing the Laplace-Beltrami
operator. This problem is strictly related to the problem of defining
intrinsically a volume in sub-Riemannian geometry. Indeed starting
from dimension 5, the Popp volume (that is a possible generalization
of the Riemannian volume) does not cincide anymore with the spherical
Hausdorff volume. The random-walk point of view provides some
interesting information.