Integration of nonsmooth $2$-forms: from Young to Itô and Stratonovich

Abstract

We show that geometric integrals of the type ${\int }_{\Omega }fd{g}^1\wedge d{g}^2$ can be defined over a two-dimensional domain $\Omega$ when the functions $f$, $g^1$, $g^2:{\mathbb{R}}^2\to \mathbb{R}$ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of $\Omega$ has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R.~Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder 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.