Lyapunov's Second Method for Random Dynamical Systems

The method of Lyapunov functions (Lyapunov's second or direct method) was originally developed for studying the stability of a xed point of an autonomous or non-autonomous diierential equation. It was then extended from xed points to sets, from diierential equations to dynamical systems and to stochastic diierential equations. We go one step further and develop Lyapunov's 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.