Spectral Theory for Random Dynamical Systems

In this chapter, we introduce two spectral theories for linear random dynamical systems. The first one is the Lyapunov spectrum which is also known as the multiplicative ergodic theorem. The second one is the dichotomy spectrum (also known as Sacker-Sell spectrum) defined in terms of exponential dichotomy. We also establish explicit formulas for Lyapunov exponent of planar bilinear stochastic differential equations. The materials of this chapter are taken from Anh and Doan (Stoch Analy Appl 35(4):662–676, 2017) and Callaway et al. (Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53:1548–1574, 2017).