Random dynamical system

Summary

A random dynamical system is a dynamical system whose rule of evolution is chosen at random at each time. The randomness comes from a noise that is itself a measure-preserving dynamical system. The evolution is a cocycle over the noise. The state after time $s + t$ is obtained by first evolving for time $s$ and then evolving for time $t$ with the shifted noise. Stochastic differential equations and random iterations of maps give random dynamical systems. The framework allows notions of deterministic dynamics, such as attractors, invariant measures and Lyapunov exponents, to be used for systems with noise.