Rough center manifolds

Abstract

Since the breakthrough in rough paths theory for stochastic ordinary differential equations (SDEs), there has been a strong interest in investigating the rough differential equation (RDE) approach and its numerous applications. Rough path techniques can stay closer to deterministic analytical methods and have the potential to transfer many pathwise ordinary differential equation (ODE) techniques more directly to a stochastic setting. However, there are few works that analyze dynamical properties of RDEs and connect the rough path / regularity structures, ODE and random dynamical systems approaches. Here we contribute to this aspect and analyze invariant manifolds for RDEs. By means of a suitably discretized Lyapunov-Perron-type method we prove the existence and regularity of local center manifolds for such systems. Our method directly works with the RDE and we exploit rough paths estimates to obtain the relevant contraction properties of the Lyapunov-Perron map.

arXiv commentTo appear in SIAM Journal on Mathematical Analysis

References in corpus (5)

• Normal form transforms separate slow and fast modes in stochastic dynamical systems
• The dynamical sine-Gordon model
• Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
• Rough Burgers-like equations with multiplicative noise
• The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations

Cited by in corpus (2)

• Reduction Methods in Climate Dynamics -- A Brief Review
• A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds