Itô-Type Stochastic Parabolic Partial Differential Equations in Hilbert Spaces: Stability and Convergence Results via Lyapunov-Like Functions

In this article, a set of sufficient conditions for the stability and convergence of the solution process of the It -type stochastic parabolic partial differential equation (SPPDE) are established by developing a comparison principle in the context of Lyapunov-like functions coupled with differential inequalities. Here 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, an effort has been made to obtain sharper sufficient conditions via Lyapunov techniques and comparison principle. This comparison technique allows one to derive the stability and convergence properties of the nonlinear SPPDE by deriving the stability and convergence properties of a corresponding differential equation without calculating the solution process of the SPPDE explicitly. An example is given to illustrate the significance of the developed results.