A Non-Smooth Stochastic Lyapunov Function and Its Relationship With Viscosity Solutions

In this paper, we derive sufficient conditions that the origin of a stochastic system is stable in some meanings by a particular shape of a stochastic Lyapunov function without smoothness at some points. The treated stability properties are transient stability and transient asymptotic stability for the origins being non-equilibria, and stability in probability and asymptotic stability in probability for the origins being equilibria. Furthermore, we also discuss the relationship between our stochastic Lyapunov functions and viscosity supersolutions, which are the notion often used for analyzing non-smooth Lyapunov functions for deterministic systems. Then, we propose a relaxed notion of viscosity weak supersolution for our stochastic Lyapunov function.