Young Differential Equations with Power Type Nonlinearities

Abstract

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