Calculus of Variations and Geometric Measure Theory

G. Alberti - E. Stepanov - D. Trevisan

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

created by stepanov on 12 Dec 2019
modified on 13 Oct 2023


Accepted Paper

Inserted: 12 dec 2019
Last Updated: 13 oct 2023

Journal: J. Funct. Anal.
Year: 2019


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 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.

Keywords: Young integral, Rough paths, Sewing lemma, Itô integral, Stratonovich integral