Theory and application of stability for stochastic reaction diffusion systems

So far, the Lyapunov direct method is still the most effective technique in the study of stability for ordinary differential equations and stochastic differential equations. Due to the shortage of the corresponding Itô formula, this useful method has not been popularized in stochastic partial differential equations. The aim of this work is to try to extend the Lyapunov direct method to the Itô stochastic reaction diffusion systems and to establish the corresponding Lyapunov stability theory, including stability in probablity, asymptotic stability in probability, and exponential stability in mean square. As the application of the obtained theorems, this paper addresses the stability of the Hopfield neural network and points out that the main results obtained by Holden Helge and Liao Xiaoxin et al. can be all regarded as the corollaries of the theorems presented in this paper.