Decay Rate of Iterated Integrals of Branched Rough Paths

TL;DR

This paper proves a factorial decay estimate for iterated integrals of non-geometric rough paths, extending Lyons' 1994 results and addressing limitations of previous methods with a new concavity approach.

Contribution

It establishes a factorial decay bound for non-geometric rough paths, confirming a conjecture by Gubinelli and introducing novel analytical techniques.

Findings

Proved factorial decay estimate for non-geometric rough paths

Identified limitations of the neoclassical inequality in this context

Extended Lyons' proof to a broader class of integrals

Abstract

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.