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Based on the recent development of the framework of Volterra rough paths, weconsider here the probabilistic construction of the Volterra rough pathassociated to the fractional Brownian motion with $H>\frac{1}{2}$ and for thestandard Brownian motion. The Volterra kernel $k(t,s)$ is allowed to besingular, 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 basedon a modified Garsia-Rodemich-Romsey lemma which has an interest in its ownright, as well as tools from Malliavin calculus. A discussion of challenges andpotential extensions is provided.

Volterra equations driven by rough signals 3: Probabilistic construction of the Volterra rough path for fractional Brownian motions