Planarly branched rough paths and rough differential equations on homogeneous spaces

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 Hölder index of the driving path is equal to one, we show convergence of the planar forest expansion in a small time interval.

arXiv comment37 pages, uses axodraw. This version accepted for publication in J. Differential Equations

References in corpus (19)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Homology of generalized partition posets
• Two interacting Hopf algebras of trees
• On post-Lie algebras, Lie--Butcher series and moving frames
• Quasi-shuffle products
• Runge-Kutta methods and renormalization
• On the Lie enveloping algebra of a post-Lie algebra
• Controlling Rough Paths
• Monomial bases for free pre-Lie algebras
• The Magnus expansion, trees and Knuth's rotation correspondence
• Decay Rate of Iterated Integrals of Branched Rough Paths
• On an extension of Knuth's rotation correspondence to reduced planar trees
• A comodule-bialgebra structure for word-series substitution and mould composition
• Tree algebras over topological vector spaces in rough path theory
• Lie groups of controlled characters of combinatorial Hopf algebras
• Combinatorial Hopf algebra for interconnected nonlinear systems
• Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
• Convergence of Lie group integrators
• Lie-Butcher series, Geometry, Algebra and Computation

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• An introduction to infinite-dimensional differential geometry
• Post-symmetric braces and integration of post-Lie algebras
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• A Survey on the Munthe-Kaas-Wright Hopf Algebra
• On the definition of a solution to a rough differential equation
• Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
• Translations of rough paths in combinatorial Hopf algebras
• Combinatorial Hopf algebras from restriction species with preorder cuts
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
• Flows driven by multi-indices Rough Paths