Pathwise unstable invariant manifolds reduction for stochastic evolution equations driven by nonlinear noise

Abstract

This paper is concerned with the pathwise dynamics of the stochastic evolution equation: du + Audt = F(u)dt + G(u)dW(t) on a separable Hilbert space H with the Lipschitz continuous drift term F(u) as well as the Lipschitz continuous diffusion term G(u). We first introduce the notion of generalized random dynamical systems (GRDSs) and show that the equation can generate a GRDS. We then construct a pathwise unstable manifold for the GRDS provided that the Lipschitz constants of the drift term and the diffusion term satisfy a spectral gap condition. At last, we present a pathwise unstable manifold reduction for the GRDS.