ABSTRACT
The purpose of this chapter is to investigate stability of nonlinear stochastic differential equations. We first formulate this problem as a comparison of the quadratic functionals of two stochastic differential equations. Then, coercivity types of criteria, actually, Lyapunov’s function methods are presented to deal with stability properties for strong and mild solutions. A Lyapunov function programme is also carried out to handle, apart from stability, ultimate boundedness and associated existence and uniqueness of invariant measures. We investigate the decay rate of mild solutions for a class of nonlinear stochastic evolution equations. Lastly, based on the viewpoint of perturbations of infinite dimensional deterministic systems, the so-called stabilization by white noise sources of systems is studied.
