We establish two results concerning a class of geometric rough paths X which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for X in α-Hölder rough path topology for all α ∈ (0, 1/2), which proves a conjecture of Friz–Victoir [13]. The second is a Hörmander-type theorem for the existence of a density of a rough differential equation driven by X, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.
