ABSTRACT
The purpose of this chapter is to establish the stability of systems defined by stochastic linear evolution equations. We mainly concern ourselves with exploring some characteristic results which are a natural extension of Lyapunov’s classical work in finite dimensional spaces. We begin our statements with the deterministic case in which a linear unbounded operator satisfying appropriate conditions generates a stable C0-semigroup of bounded linear operators. Under suitable circumstances, the characterization of mean square exponential stability is established and applied at the end of the chapter to various stochastic (partial or delay) differential equations. Subsequently, we shall establish almost sure pathwise stability of stochastic systems, a case which can be most closely related to their deterministic counterparts. In some sense, it is this kind of stability that one really likes to have in practical situations.
