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Smooth rough paths, their geometry and algebraic renormalization

Carlo Bellingeri, Peter K. Friz, Sylvie Paycha, Rosa Preiß

TL;DR

A Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on, and extensions to the quasi-geometric and the more general Hopf algebraic setting are explored.

Abstract

We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.

Smooth rough paths, their geometry and algebraic renormalization

TL;DR

A Maurer–Cartan perspective is the key to a purely algebraic form of Lyons’ extension theorem, the renormalization of rough paths following up on, and extensions to the quasi-geometric and the more general Hopf algebraic setting are explored.

Abstract

We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons extension theorem, the renormalization of rough paths in the spirit of [Bruned, Chevyrev, Friz, Preiß, A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019] as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.
Paper Structure (15 sections, 29 theorems, 175 equations)

This paper contains 15 sections, 29 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Smooth geometric rough paths
  3. Definitions and fundamental properties
  4. Canonical sum and minimal coupling of smooth geometric rough models
  5. Differential equations driven by SGRP
  6. Algebraic renormalization of SGRP
  7. Smooth quasi-geometric rough paths
  8. Quasi-shuffle algebras
  9. Smooth quasi-geometric rough paths
  10. Differential equations driven by SQGRP
  11. Algebraic renormalisation of SQGRP
  12. Recast in an abstract Hopf algebra framework
  13. Smooth rough paths on a Hopf algebra
  14. Translation of smooth rough paths on a Hopf algebra
  15. Canonical sum and minimal coupling of smooth rough paths

Key Result

Proposition 2.6

Given a sgrm $\mathbf{X}$ over $\mathbb{R}^d$, for all times $s$ one has, The analogue statement holds for $N$-sgrm's, with ${{\mathcal{L}}}((\mathbb{R}^d))$ replaced by its truncation $\mathcal{L}^N(\mathbb{R}^d)$. We call the path in prop:diagonal_derivative the diagonal derivative of $\mathbf{X}$.

Theorems & Definitions (95)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8: Fundamental Theorem of sgrm
  • proof
  • ...and 85 more