Long-Time Stability Analysis for Stochastic Evolution Equations with Multiplicative Noise

TL;DR

This paper analyzes the long-term stability of linear stochastic evolution equations with multiplicative noise, providing explicit conditions for stability and demonstrating the preservation of these properties through numerical schemes.

Contribution

It introduces explicit stability criteria for stochastic evolution equations with multiplicative noise and shows their preservation in numerical discretizations.

Findings

Established explicit conditions for p-th moment and almost sure exponential stability.

Demonstrated stability preservation in spectral Galerkin and Euler–Maruyama schemes.

Numerical simulations confirm theoretical stability results.

Abstract

In this paper, we study the long-time stability behavior of a class of linear stochastic evolution equations in a Hilbert space with multiplicative noise. Explicit sufficient conditions for $p$-th moment and almost sure exponential stability are established, highlighting the interplay between the principal eigenvalue of the governing operator, the drift coefficient, and the noise intensity. The relationship between these two notions of stability is also clarified. Applications to several stochastic partial differential equations are presented. In addition, a fully discrete spectral Galerkin method together with the implicit Euler--Maruyama scheme is shown to preserve these stability properties at the discrete level. Finally, numerical simulations are provided to confirm the theoretical results.