Geometric aspects of Young Integral: decomposition of flows

TL;DR

This paper explores the geometric properties of Young differential equations driven by Hölder continuous paths, introducing new geometrical constructions and a flow decomposition based on diffeomorphisms.

Contribution

It develops a geometric framework for Young differential equations, including formulas and concepts like parallel transport and covariant derivatives, and provides a flow decomposition theorem.

Findings

Established Young Itô geometrical formula

Constructed horizontal lift and parallel transport in this context

Proved a flow decomposition theorem for Young differential equations

Abstract

In this paper we study geometric aspects of dynamics generated by Young differential equations (YDE) driven by $\alpha$-H\"older trajectories with $\alpha \in (1/2, 1)$. We present a number of properties and geometrical constructions on this low regularity context: Young It\^o geometrical formula, horizontal lift in principal fibre bundles, parallel transport, covariant derivative, development and anti-development, among others. Our main application here is a geometrical decomposition of flows generated by YDEs according to diffeomorphisms generated by complementary distributions (integrable or not). The proof of existence of this decomposition is based on an Young It\^o-Kunita formula for $\alpha$-H{\"o}lder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022).