A note on higher dimensional $p$-variation

TL;DR

This paper clarifies two related notions of $p$-variation for functions on $[0,T]^2$, demonstrating their near equivalence and relevance for Gaussian rough paths analysis.

Contribution

It establishes that two slightly different definitions of $p$-variation are essentially equivalent, simplifying their application in Gaussian rough path theory.

Findings

The two notions of $p$-variation are $ ext{ extasciitilde}$ extbackslash epsilon-close.

Arguments for Gaussian rough paths apply to both notions with minor adjustments.

Clarifies previous ambiguities in $p$-variation concepts for functions on $[0,T]^2$.

Abstract

We discuss $p$-variation regularity of real-valued functions defined on $[0,T]^2$, based on rectangular increments. When $p>1$, there are two slightly different notions of $p$-variation; both of which are useful in the context of Gaussian rough paths. Unfortunately, these concepts were blurred in previous works; the purpose of this note is to show that the aforementioned notions of $p$-variations are "$\epsilon$-close". In particular, all arguments relevant for Gaussian rough paths go through with minor notational changes.