Stationary Distributions of Second Order Stochastic Evolution Equations with Memory in Hilbert Spaces

TL;DR

This paper investigates the stationary distributions of second-order stochastic evolution equations with memory in Hilbert spaces, using reduction techniques and asymptotic analysis to understand their long-term behavior.

Contribution

It introduces a novel approach to analyze stationarity of second-order stochastic equations with memory by reducing them to first-order systems and applying spectral theorems.

Findings

Established conditions for stationarity of the systems.

Derived asymptotic behavior of dissipative second-order equations.

Provided an example with stochastic delay wave equations.

Abstract

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Levy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart-Pruss-Greiner's theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, an example of stochastic damped delay wave equations with memory is presented to illustrate our theory.