On Uniformly Subelliptic Operators and Stochastic Area

Abstract

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.