Center Manifolds for Rough Partial Differential Equations

TL;DR

This paper establishes a center manifold theorem for rough PDEs driven by nonlinear multiplicative noise, extending the theory of random dynamical systems to include such stochastic partial differential equations.

Contribution

It introduces a novel center manifold theorem for rough PDEs using rough path theory and Lyapunov-Perron methods, applicable to reaction-diffusion and Swift-Hohenberg equations.

Findings

Proves the existence of a random center manifold for rough PDEs.

Applies the theorem to reaction-diffusion and Swift-Hohenberg equations.

Develops a new analytical framework combining rough paths and semigroup theory.

Abstract

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a $\gamma$-H\"older rough path, for $\gamma\in(1/3,1/2]$. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.