Stability of Stochastic Functional Differential Equations

In this chapter, we shall investigate stability properties of stochastic functional differential equations in infinite dimensions. We begin with an argument of reducing the stability problem of retarded functional linear deterministic equations to a class of C0-semigroups of bounded linear operators so as to find exact regions of stability. The characteristic conditions of mean square exponential stability established in Chapter 2 for linear equations are extended to a class of stochastic linear functional equations with time lag. A kind of coercive condition is formulated to secure desired decay behavior of strong solutions for nonlinear stochastic functional differential equations. The methods of Lyapunov and Razumikhin functionals are emphasized and contrasted to describe stability properties of mild solutions for semilinear stochastic evolution equations with memory.