On Young Systems

TL;DR

This paper investigates differential equations driven by paths with bounded p-variation, focusing on Young systems, and develops formulas and conditions for their solutions, symmetries, and conserved quantities.

Contribution

It introduces new formulas for Young integrals, flow compositions, and conditions for symmetries and conservation laws in Young systems.

Findings

Derived an Itô-Kunita-Ventzel type formula for Young integrals.

Established necessary conditions for conserved quantities and symmetries.

Analyzed the Cauchy problem for Young partial differential equations.

Abstract

In this article, we study differential equations driven by continuous paths with with bounded $p$-variation for $1 \leq p< 2$ (Young systems). The most important class of examples of theses equations is given by stochastic differential equations driven by fractional Brownian motion with Hurst index $H >\frac{1}{2}$. We give a formula type It\^o-Kunita-Ventzel and a substitution formula adapted to Young integral. It allows us to give necessary conditions for existence of conserved quantities and symmetries of Young systems. We give a formula for the composition of two flows associated to Young sistems and study the Cauchy problem for Young partial differential equations.