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

Abstract

In this survey we recall the results obtained in [Ungureanu,Electronic Journal of Qualitative Theory of Differential Equations, 2004] where we gave a representation theorem for the solutions of stochastic differential equations in Hilbert spaces. Using this representation theorem we obtained deterministic characterizations of exponential stability and uniform observability in [Ungureanu,Electronic Journal of Qualitative Theory of Differential Equations, 2004], [Ungureanu, Operator Theory: Advances and Applications, Birkhauser Verlag Basel, 2005] and we will prove a result of Datko type concerning the exponential dichotomy of stochastic equations.

Keywords

• Lyapunov equations
• Stochastic differential equations
• Uniform exponential stability
• Uniform observability
• Uniform exponential dichotomy