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

TL;DR

This paper advances the theory of probabilistic rough paths by establishing key properties of random controlled rough paths, including stability of rough integration and composition, using Lions-Taylor expansions within a complex algebraic framework.

Contribution

It introduces a rigorous Lions-Taylor expansion for probabilistic rough paths and proves stability of rough integration and composition operators in this setting.

Findings

Established closedness of rough integration operator

Proved stability of composition with smooth functions

Developed a higher-order Lions-Taylor expansion

Abstract

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.