Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

TL;DR

This paper develops a framework for invariant manifolds in ill-posed stochastic evolution equations with nonlinear noise, introducing new methods to handle the lack of classical conditions and establishing the existence of stable and unstable manifolds.

Contribution

It introduces a novel approach to construct mean-square invariant manifolds for ill-posed stochastic equations without the Hille-Yosida condition, using a modified variation of constants and Lyapunov-Perron method.

Findings

Existence of mean-square random unstable invariant manifold.

Existence of mean-square stable invariant set.

New techniques for ill-posed stochastic equations with nonlinear noise.

Abstract

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.