Lyapunovʹs Second Method for Random Dynamical Systems

The method of Lyapunov functions (Lyapunovʹs second or direct method) was originally developed for studying the stability of a fixed point of an autonomous or non-autonomous differential equation. It was then extended from fixed points to sets, from differential equations to dynamical systems and to stochastic differential equations. We go one step further and develop Lyapunovʹs second method for random dynamical systems and random sets, together with matching notions of attraction and stability. As a consequence, Lyapunov functions will also be random. Our test is that the extension be coherent in the sense that it reduces to the deterministic theory in case the noise is absent, and that we can prove that a random set is asymptotically stable if and only if it has a Lyapunov function. Several examples are treated, including the stochastic Lorenz system.