Invariant Manifolds for Random Parabolic Evolution Equations with almost sectorial operators

TL;DR

This paper develops a framework for analyzing the random dynamics of stochastic evolution equations with non-dense domains, including those with white noise boundary conditions, by establishing invariant manifolds.

Contribution

It introduces a novel approach combining integrated semigroup theory and invariant manifold theory for stochastic evolution equations with non-dense domains.

Findings

Existence of stable, unstable, and center manifolds around stationary trajectories.

Application to stochastic parabolic equations with boundary white noise.

Framework applicable to various evolution equations with non-homogeneous boundary conditions.

Abstract

In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with non-homogeneous boundary conditions of white noise type. We prove the existence of stable, unstable, and center manifolds around a stationary trajectory by combining integrated semigroup theory and invariant manifold theory. The results are applied to stochastic parabolic equations with white noise at the boundary.