Unstable manifolds for rough evolution equations

TL;DR

This paper establishes the existence of global solutions and local unstable manifolds for rough evolution equations driven by finite-dimensional Hölder rough paths, advancing the understanding of their dynamical properties.

Contribution

It introduces a framework for solving rough evolution equations driven by Hölder rough paths and constructs local unstable manifolds using a discretized Lyapunov-Perron method.

Findings

Proved global-in-time solutions for rough evolution equations.

Established the existence of local unstable manifolds.

Generated random dynamical systems from solutions.

Abstract

In this paper, we consider a class of evolution equations driven by finite-dimensional $\gamma$-H\"{o}lder rough paths, where $\gamma\in(1/3,1/2]$. We prove the global-in-time solutions of rough evolution equations(REEs) in a sutiable space, also obtain that the solutions generate random dynamical systems. Meanwhile, we derive the existence of local unstable manifolds for such equations by a properly discretized Lyapunov-Perron method.