ABSTRACT
In this chapter, we introduce the definitions of principal spectrum and principal Lyapunov exponents and exponential separation for a family of general parabolic equations and present their basic properties. We also present a multiplicative ergodic theorem for a family of general parabolic equations. This chapter is organized as follows. In Section 3.1 we introduce the definitions of principal spectrum and Lyapunov exponents of (2.0.1)+(2.0.2) and study their basic properties. We introduce the definition of exponential separation and investigate relevant basic properties in Section 3.2. The existence of exponential separation is explored in Section 3.3. In Section 3.4 we present a multiplicative ergodic theorem. Special properties for a family of general smooth parabolic equations are discussed in Section 3.5. Some remarks on parabolic equations in nondivergence form are given in Section 3.6. This chapter ends up with an appendix on parabolic equations on one-dimensional space domain.
