Stationary Solutions of Neutral Stochastic Partial Differential Equations with Delays in the Highest-Order Derivatives

TL;DR

This paper investigates the existence and uniqueness of stationary solutions for a class of neutral stochastic partial differential equations with delays in derivatives, focusing on distributed delays and providing an illustrative example.

Contribution

It introduces new results on stationary solutions for neutral SPDEs with delays in highest-order derivatives, including delays in spatial and temporal terms.

Findings

Established conditions for existence and uniqueness of stationary solutions

Analyzed the impact of distributed delays on system stability

Provided an illustrative example demonstrating the theory

Abstract

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analysed to explain the theory in this work.