Unstable manifolds for rough evolution equations

Abstract

In this paper, we consider a class of rough nonlinear evolution equations driven by infinite-dimensional $\gamma$-Hölder rough paths with $\mathrm{\gamma }\mathrm{\in }(1/3,1/2]$. First, we give a proper integral with respect to infinite-dimensional $\gamma$-Hölder rough paths by using rough paths theory. Second, we obtain the global in time solution and random dynamical system of rough evolution equation. Finally, we derive the existence of local unstable manifolds for rough evolution equations by a properly discretized Lyapunov–Perron method.