Planarly branched rough paths and rough differential equations on homogeneous spaces

TL;DR

This paper develops a framework for solving rough differential equations on homogeneous spaces using planarly branched rough paths, extending existing theories to a more general geometric setting with explicit solution expansions.

Contribution

It introduces planarly branched rough paths based on planar rooted forests and applies them to rough differential equations on homogeneous spaces, including an extension theorem and convergence results.

Findings

Defined planarly branched rough paths using Munthe-Kaas--Wright Hopf algebra.

Established an extension theorem analogous to Lyons' theorem.

Proved convergence of the forest expansion under certain conditions.

Abstract

The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar forests. For this we develop the notion of planarly branched rough paths, following M. Gubinelli's branched rough paths. The definition is similar to the one in the flat case, the main difference being the replacement of the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We show how the latter permits to handle rough differential equations on homogeneous spaces using planarly branched rough paths, the same way branched rough paths are used in the context of rough differential equations on finite-dimensional vector spaces. An analogue of T. Lyons' extension theorem is proven. Finally, under analyticity assumptions on the coefficients and when the H\"older index of the driving path is equal to one, we show convergence of the planar forest expansion in a small time interval.