Representation Theorem for Stochastic Differential Equations in Hilbert Spaces and Its Applications

In this survey we recall the results obtained in [16] where we gave a representation theorem for the solutions of stochastic di¤erential equations in Hilbert spaces. Using this representation theorem and the deterministic characterizations of exponential stability and uniform observability obtained in [16], [17], we will prove a result of Datko type concerning the exponential dichotomy of stochastic equations. 1 Introduction In [16] V. Ungureanu established a representation theorem (see Theorem 3) for the mild solutions of linear stochastic di¤erential equations. More precisely, in [16] a Lyapunov equation is associated to the discussed linear stochastic di¤erential equation and it is established a relation between the mean square of the mild solution of the stochatic equation and the mild solution of the Lyapunov equation. This representation theorem is a powerful tool which allow us to obtain deterministic characterizations of di¤erent properties of solutions of linear di¤erential stochastic equations. The aim of this survey is to illustrate how problems like uniform exponential stability, uniform observability or uniform exponential dichotomy of stochastic equations can be solved by using the result obtained in [16]. The survey is organized as it follows. In the second section we recall basic facts concerning linear stochastic di¤erential equations and Lyapunov equations, which we need in the sequel. The representation theorem is stated in the third section. In section 4 we introduce a solution operator associated to the Lyapunov equation associated to the stochastic di¤erential equation and we establish some of its 2000 Mathematics Subject Classi…cation: 93E15, 34D09, 93B07.