Planarly branched rough paths and rough differential equations on homogeneous spaces

Curry, Charles; Ebrahimi‐Fard, Kurusch; Manchon, Dominique; Munthe-Kaas, Hans

doi:10.1016/j.jde.2020.06.058

Cited by 18 publications

(24 citation statements)

References 28 publications

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“…The following result has already been proved in the case where the underlying Hopf algebra

scriptH

is combinatorial by Curry, Ebrahimi‐Fard, Manchon and Munthe‐Kaas in [, Theorem 4.3]. We remark that their proof works without modifications in our context so we have Theorem Let

X : false[0, 1 false] \to G^{N}

be a

γ

‐Hölder path with

{double-struckX}_{0} = 1

and suppose that

γ^{- 1} \notin N

.…”

Section: Construction Of Rough Pathsmentioning

confidence: 68%

“…Remark Given the level of generality in which Theorem is developed, our results also apply to the case when

scriptH

is a combinatorial Hopf algebra as defined in . In particular, we also have a construction theorem for planarly branched rough paths which are characters over Munthe‐Kaas and Wright's Hopf algebra of Lie group integrators .…”

Section: Applicationsmentioning

confidence: 91%

“…Remark By specializing this definition to different choices of

scriptH

we recover both GRPs where

scriptH

is the shuffle Hopf algebra over an alphabet, branched rough paths where

scriptH

is the Butcher–Connes–Kreimer Hopf algebra on decorated non‐planar rooted trees, and also planarly branched rough paths .…”

Section: Construction Of Rough Pathsmentioning

confidence: 99%

See 2 more Smart Citations

The geometry of the space of branched rough paths

Tapia

Zambotti

2020

Proc. London Math. Soc.

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We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between these two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker–Campbell–Hausdorff formula, on a constructive version of the Lyons–Victoir extension theorem and on the Hairer–Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.

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“…The following result has already been proved in the case where the underlying Hopf algebra

scriptH

is combinatorial by Curry, Ebrahimi‐Fard, Manchon and Munthe‐Kaas in [, Theorem 4.3]. We remark that their proof works without modifications in our context so we have Theorem Let

X : false[0, 1 false] \to G^{N}

be a

γ

‐Hölder path with

{double-struckX}_{0} = 1

and suppose that

γ^{- 1} \notin N

.…”

Section: Construction Of Rough Pathsmentioning

confidence: 68%

“…Remark Given the level of generality in which Theorem is developed, our results also apply to the case when

scriptH

Section: Applicationsmentioning

confidence: 91%

See 1 more Smart Citation

The geometry of the space of branched rough paths

Tapia

Zambotti

2020

Proc. London Math. Soc.

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show abstract

“…The article [45] study a Runge-Kutta scheme for RDE, while [7,20] use algebraic results on Taylor series, B-series and the Faà di Bruno formula as a core. A variant to branched rough paths consists in using planar trees [24]. While these articles focus mainly on the properties of the driving paths, the algebraic properties of the vector fields play a large role here.…”

Section: Introductionmentioning

confidence: 99%

Constructing general rough differential equations through flow approximations

Lejay

2022

Electron. J. Probab.

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The non-linear sewing lemma constructs flows of rough differential equations from a broad class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kinds of driving paths. We also extend the notion of driving rough path. We introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.

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“…al. in [8], where the concept of a rough path over any combinatorial Hopf algebra was defined. Inspired by Lie group integration theory [16] and the notion of Lie-Butcher series [18] they furthermore showed how rough paths over the Munthe-Kaas-Wright Hopf algebra of planar rooted trees can be use to define solutions to rough differential equations on homogeneous spaces.…”

Section: Introductionmentioning

confidence: 99%

Translations of rough paths in combinatorial Hopf algebras

Rahm¹

2021

Preprint

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We generalize Bruned et. al.'s notion of translation in geometric and branched rough paths to a notion of translation in rough paths over any combinatorial Hopf algebra. We show that this notion of translation is equivalent to two bialgebras being in cointeraction, subject to certain additional conditions. We argue that reformulating translations in terms of substitutions, provides simpler conditions for the cointeraction formulation. As a concrete example, we describe translations in planarly branched rough paths.

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Planarly branched rough paths and rough differential equations on homogeneous spaces

Cited by 18 publications

References 28 publications

The geometry of the space of branched rough paths

The geometry of the space of branched rough paths

Constructing general rough differential equations through flow approximations

Translations of rough paths in combinatorial Hopf algebras

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