A Lyapunov–Cycle Mesh Framework for Stochastic Stability of Positive Nonlinear Systems with Higher-Order Curvature

We propose a Lyapunov–cycle mesh framework for the stochastic stability of positive nonlinear systems with higher-order curvature dynamics. The approach combines a log-convex Lyapunov barrier with a third-order curvature subsystem and a multiplicative mirror-ascent update that preserves positivity. Stochastic effects are incorporated through an \varepsilon- bounded drift condition on a hierarchical mesh of weakly coupled subsystems. Using supermartingale arguments and uniform integrability, we establish finite expected hitting times, ultimate boundedness in probability, and L^1-convergence. The framework yields forward- invariant stability regions and scales to high-dimensional settings. Applications include stochastic approximation, reinforcement learning, and quantum dynamical systems.