Abstract
This paper develops Lyapunov and converse Lyapunov theorems for discrete-time stochastic semistable nonlinear dynamical systems expressed by Itô-type difference equations possessing a continuum of equilibria. Specifically, we provide necessary and sufficient Lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous Lyapunov function whose difference operator involves a discrete-time analog of the infinitesimal generator for continuous-time Itô dynamical systems and decreases along the dynamical system sample solution sequences satisfying an inequality involving the average distance to the set of the system equilibria.
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