Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to It\^{o} and Stratonovich

TL;DR

This paper extends the Young integral to two dimensions for Hölder continuous functions, incorporating Stratonovich and Itô types, and introduces a sewing lemma for 2D rough paths.

Contribution

It develops a 2D extension of the Young integral using geometric integrals and sewing lemma, connecting to recent work by Züst and relaxing Hölder conditions.

Findings

Defined 2D geometric integrals for Hölder functions with small boundary dimension.

Established a 2D Young integral coinciding with Züst's recent integral.

Extended sewing lemma to 2D alternating functions for rough path analysis.

Abstract

We show that geometric integrals of the type $\int_\Omega f\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $\Omega$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just H\"{o}lder continuous with sufficiently large H\"{o}lder exponents and the boundary of $\Omega$ has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or It\^{o} type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R.~Z\"{u}st. We further show that the Stratonovich-type summation allows to weaken the requirements on H\"{o}lder exponents of the map $g=(g^1,g^2)$ when $f(x)=F(x,g(x))$ with $F$ sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.