Young Differential Equations with Power Type Nonlinearities

TL;DR

This paper develops methods to construct solutions for Young differential equations with power-type nonlinearities, especially in cases where classical Young integration techniques do not apply, by extending integration methods.

Contribution

It introduces extended Young integral techniques to handle differential equations with power nonlinearities outside classical existence regimes.

Findings

Constructed solutions for equations with power nonlinearities.

Extended Young integration methods to broader parameter ranges.

Provided existence results beyond classical Young theory.

Abstract

In this note we give several methods to construct nontrivial solutions to the equation $dy_{t}=\sigma(y_{t}) \, dx_{t}$, where $x$ is a $\gamma$-H\"older $R^{d}$-valued signal with $\gamma\in(1/2,1)$ and $\sigma$ is a function behaving like a power function $|\xi|^{\kappa}$, with $\kappa\in(0,1)$. In this situation, classical Young integration techniques allow to get existence and uniqueness results whenever $\gamma(\kappa+1)>1$, while we focus on cases where $\gamma(\kappa+1)\le 1$. Our analysis then relies on some extensions of Young's integral allowing to cover the situation at hand.