A Girsanov approach to slow parameterizing manifolds in the presence of noise

Abstract

This work investigates a three-dimensional slow-fast stochastic system with quadratic nonlinearity and additive noise, inspired by fluid dynamics. The deterministic counterpart exhibits a periodic orbit and a slow manifold. We demonstrate that, under specific parameter regimes, this deterministic slow manifold can serve as an approximate parameterization of the fast variable by the slow variables within the stochastic system. Building upon this parameterization, we derive a two-dimensional reduced model, a stochastic Hopf normal form, that captures the essential dynamics of the original system. Both the original and the reduced systems possess ergodic invariant measures, characterizing their long-term behavior. We quantify the discrepancy between the original system and its slow approximation by deriving error estimates involving the Wasserstein distance between the marginals of these invariant measures along the radial component. These error bounds are shown to be controlled by a parameterization defect, which measures the quality of the fast-slow variable parameterization. A key technical innovation lies in the application of Girsanov's theorem to obtain these error estimates in the presence of oscillatory instabilities. Furthermore, we extend our analysis to regimes exhibiting an "inverted" timescale separation, where the variable to be parameterized evolves on a slower timescale than the resolved variables. To address these more challenging scenarios, we introduce path-dependent coefficients in the parameterizing manifold, enabling the derivation of robust error bounds for the corresponding reduced model. Numerical simulations complement our theoretical findings, providing insights into the model's behavior and exploring parameter regimes beyond the scope of our analytical results.