Adaptive Stochastic Contraction in Time-Varying Systems: A Unified Framework for Persistent Noise Dynamics

This work presents a preliminary framework for analyzing stochastic iterative systems with time-varying nonlinear mappings under persistent noise. We introduce an adaptive stochastic contraction perspective and derive an expectation-level convergence bound indicating stability within a noise-dependent neighborhood. The current version focuses on conceptual development and initial empirical validation across synthetic and real-world datasets. While the framework provides an intuitive and interpretable formulation, several aspects including formal probabilistic assumptions, stronger theoretical guarantees (e.g., high-probability bounds), and rigorous experimental validation are under active development. This preprint is shared to establish an early version of the idea and invite future refinement. Subsequent versions will include improved theoretical formalization, reproducible experimental protocols, and comprehensive comparisons with existing methods in stochastic approximation and contraction theory.