  In this paper we study left-invariant Laplacians on compact connected groups
that are form-comparable perturbations of bi-invariant Laplacians. Our results
show that Gaussian bounds for derivatives of heat kernels enjoyed by certain
bi-invariant Laplacians hold for their form-comparable perturbations. We
further show that the parabolic operators associated with such left-invariant
Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic
in various senses.

Probability;Analysis of PDEs