  Based on the recent development of the framework of Volterra rough paths, we
consider here the probabilistic construction of the Volterra rough path
associated to the fractional Brownian motion with $H>\frac{1}{2}$ and for the
standard Brownian motion. The Volterra kernel $k(t,s)$ is allowed to be
singular, and behaving similar to $|t-s|^{-\gamma}$ for some $\gamma\geq 0$.
The construction is done in both the Stratonovich and It\^o sense. It is based
on a modified Garsia-Rodemich-Romsey lemma which has an interest in its own
right, as well as tools from Malliavin calculus. A discussion of challenges and
potential extensions is provided.
