  Recently, Hairer--Pillai proposed the notion of $\theta$-roughness of a path
which leads to a deterministic Norris lemma. In the Gubinelli framework
(Hoelder, level 2) of rough paths, they were then able to prove a Hoermander
type result (SDEs driven by fractional Brownian motion, $H>1/3$). We take a
step back and propose a natural "roughness" condition relative to a given
$p$-rough path in the sense of Lyons; the aim being a Doob-Meyer result for
rough integrals in the sense of Lyons. The interest in our (weaker) condition
is that it is immediately verified for large classes of Gaussian processes,
also in infinite dimensions. We conclude with an application to non-Markovian
system under Hoermander's condition.
