Stability and Convergence Results for Itô-Type Parabolic Partial Differential Equations in Hilbert Spaces

Abstract

In this article, we consider the following Itô-type stochastic parabolic partial differential equation where A and C are families of nonlinear operators in Hilbert spaces and w t is a Hilbert valued Wiener process. Under certain regularity conditions of operators A and C to guarantee a strong solution process of this partial differential equation, the concept of vector Lyapunov-like functional technique coupled with partial differential inequalities are utilized to develop a comparison principle to investigate various types of stability and convergence in the pth moment and in probability of the solution process of the system. An example is provided to demonstrate the significance of the results developed in this article.