  We consider uniformly subelliptic operators on certain unimodular Lie groups
of polynomial growth. It was shown by Saloff-Coste and Stroock that classical
results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then
observed by Sturm that many proofs extend naturally to the setting of locally
compact Dirichlet spaces. We relate these results to what is known as rough
path theory by showing that they provide a natural and powerful analytic
machinery for construction and study of (random) geometric Hoelder rough paths.
(In particular, we obtain a simple construction of the Lyons-Stoica stochastic
area for a diffusion process with uniformly elliptic generator in divergence
form.) Our approach then enables us to establish a number of far-reaching
generalizations of classical theorems in diffusion theory including Wong-Zakai
approximations, Freidlin-Wentzell sample path large deviations and the
Stroock-Varadhan support theorem. The latter was conjectured by T. Lyons in his
recent St. Flour lecture.
