Invariant manifolds for stochastic delayed partial differential equations of parabolic type

TL;DR

This paper establishes the existence and smoothness of stable and unstable invariant manifolds for stochastic delayed parabolic PDEs, extending dynamical systems theory to stochastic PDEs with delay.

Contribution

It introduces a novel approach transforming stochastic delayed PDEs into a form suitable for invariant manifold analysis, proving their smoothness under spectral gap conditions.

Findings

Existence of Lipschitz continuous invariant manifolds

Smoothness of invariant manifolds under spectral gap condition

Application to a stochastic population model

Abstract

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.