DICHOTOMY SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

We investigate a concept of dichotomy spectrum for nonautonomous linear stochastic differential equations, which is defined with sample-wise exponential dichotomy of the two-parameter flow generated by the equation. We use random norm and cohomology to capture the nature of the stochastic nonuniformity. The main result is our spectral theorem stating that the dichotomy spectrum consists of compact random intervals with corresponding spectral manifolds, which are Oseledets spaces if the equation generates a random dynamical system. The dichotomy spectrum is nonrandom and equals the Lyapunov spectrum if the stochastic differential equation is Lyapunov regular.