H\"older-continuous rough paths by Fourier normal ordering

TL;DR

This paper introduces a novel Fourier normal ordering method to explicitly construct geometric rough paths over paths with finite 1/α-variation, utilizing Hopf algebra structures and Besov norms for H"older continuity.

Contribution

It presents a new regularization technique called Fourier normal ordering for constructing rough paths, connecting combinatorics of decorated trees with algebraic structures.

Findings

Constructs explicit geometric rough paths for paths with finite 1/α-variation.

Uses Fourier normal ordering to regularize iterated integrals.

Proves H"older continuity using Besov norms.

Abstract

We construct in this article an explicit geometric rough path over arbitrary $d$-dimensional paths with finite $1/\alpha$-variation for any $\alpha\in(0,1)$. The method may be coined as 'Fourier normal ordering', since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest Fourier frequencies. In doing so, there appear non-trivial tree combinatorics, which are best understood by using the structure of the Hopf algebra of decorated rooted trees (in connection with the Chen or multiplicative property) and of the Hopf shuffle algebra (in connection with the shuffle or geometric property). H\"older continuity is proved by using Besov norms. The method is well-suited in particular in view of applications to probability theory (see the companion article \cite{Unt09} for the construction of a rough path over multidimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or \cite{Unt09ter} for a short survey in that case).