Exponential stability of stochastic systems: A pathwise approach

We provide a pathwise approach using semigroup technique to study the asymptotic stability for stochastic differential equations which admit a unique equilibrium. The driving noises in consideration are [Formula: see text] — Hölder continuous with [Formula: see text], so that the perturbed systems can be solved using rough path theory, where the rough integrals are interpreted in the Gubinelli sense for controlled rough paths. Our approach suggests an alternative method for stochastic systems with standard Brownian noises, by not using Itô formula but a relaxed isometry property of Itô stochastic integrals.