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Banach fixed point and flow approach for rough analysis

Yvain Bruned, Yingtong Hou, Paul Laubie, Zhicheng Zhu

Abstract

In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.

Banach fixed point and flow approach for rough analysis

Abstract

In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.
Paper Structure (9 sections, 13 theorems, 70 equations, 1 table)

This paper contains 9 sections, 13 theorems, 70 equations, 1 table.

Table of Contents

  1. Introduction
  2. Rough paths and Hopf algebras
  3. Rough paths
  4. Butcher-Connes-Kreimer Hopf algebra and branched rough paths
  5. B-series representation of cocycle-RDE solutions
  6. 1-cocycle Hopf algebra and $\mathcal{H}_{\textnormal{\tiny BCK}}$
  7. B-series solutions and controlled rough paths
  8. log-ODE solution theory of cocycle-RDEs
  9. No algebraic fixed point for multi-indices

Key Result

Theorem 1.1

Assumption assumption_fixed_point implies Assumption flow_condition.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Rough paths
  • Remark 2.2
  • example 1
  • Definition 2.3: $\mathcal{H}_\textnormal{\tiny BCK}$ perfect pairing
  • Definition 2.4: Grafting product $\triangleright_{\textnormal{\tiny BCK}}$
  • example 2: Grafting product $\triangleright_{\textnormal{\tiny BCK}}$
  • ...and 25 more