Stochastic Lyapunov functions without differentiability at supposed equilibria

Stochastic calculus indicates that allowing a Lyapunov candidate function to be non-differentiable at the origin helps its expected value to decrease. To utilize this observation, this paper investigates stability of stochastic systems, supposing that the origin of the state space is an equilibrium before influenced by Gaussian white noises. For identifying such stability properties, notions of instantaneous points and almost sure equilibria, which are mutually exclusive, are introduced. This paper clarifies the relationship between the stability properties, and develops their Lyapunov-type characterizations without differentiability at the origin. Discussions are given in relation to the notion of noise-to-state stability, and this paper proposes a method to address both transient and global-in-time properties simultaneously with a single positive definite function even when stochastic noises prevent the origin from being an equilibrium.