Probabilistic rough paths II Lions-Taylor expansions and Random controlled rough paths

In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the structure of which is indexed by Lions forests. These are statistical distributions over the space of paths described by the combination of a jet on the underlying probabilistic rough path and a remainder term. The regularity of the latter facilitates the definition of a rough integral. We establish closedness and stability of two key operators on random controlled rough paths: rough integration and composition by a smooth function on the Wasserstein space. These are important results towards a complete theory of rough McKean-Vlasov equations that is still in gestation. The proof goes through a higher-order Taylor expansion for the Lions derivative which we rigorously expound. The coupled Hopf algebra structure (see \cite{salkeld2021Probabilistic}) and the Lions-Taylor expansion (established in Section \ref{section:TaylorExpansions}) introduce a number of additional challenges which mean these results are not simply a natural extension of classical theory. We dedicate this work to pursuing these details.