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

TL;DR

This work develops a rigorous framework for reducing a slow–fast stochastic system, inspired by fluid dynamics, via both deterministic slow manifolds and stochastic parameterizing manifolds (SPMs). It introduces a Girsanov-based coupling to compare long-term statistics and derives Wasserstein-type error bounds that tie the reduction quality to a parameterization defect, even in regimes without strict timescale separation or with oscillatory instabilities. The results include two main theorems for deterministic slow manifolds and their stochastic extensions, plus numerical evidence validating the theory and highlighting regimes where SPMs outperform deterministic reductions, especially under inverted timescale separation. Overall, the study provides a unified probabilistic approach to quantify and improve reduced-order models for noisy slow–fast systems with potential applications to fluid dynamics and related stochastic multiscale problems.

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.

Figures (6)

• Figure 1: Scatter plots. In Case I, the normalized parameterization defect for the slow manifold is given by $Q=1.789 \times 10^{-1}$, while in Case II, $Q=5.834 \times 10^{-1}$.
• Figure 2: Modeling skills using the slow manifold $h(x,y)=x^2+y^2$. Here for Case I, see Table \ref{['Table_CaseI']}.
• Figure 3: Modeling skills using the slow manifold $h(x,y)=x^2+y^2$. Here for Case II, see Table \ref{['Table_CaseI']}.
• Figure 4: Solution to Eq. \ref{['Eq_3DHopf_polar_base_stoch']} for Case III. Here, the variable $z(t)$ to parameterize evolves on a slower timescale than $x(t)$ and $y(t)$, a completely reverse situation compared to Cases I and II considered above.
• Figure 5: Capturing system variability. This figure depicts three snapshots of the Stochastic Parameterizing Manifold (SPM), $(h_{\tau}(M_t, \cdot))_{t\geq0}$ (with $M_t$ defined in \ref{['Eq_def_OU']}), represented by cyan, blue, and red curves. For comparison, the deterministic slow manifold is shown in yellow. Driven by the stochastic process $M_t$, the SPM dynamically evolves, effectively capturing a significant portion of the state space occupied by the full system, which exhibits substantial variance along the $r$-direction.

Theorems & Definitions (48)

• Proposition 2.1
• proof
• Remark 2.1
• Remark 2.2
• Theorem 2.2
• Remark 2.3
• Lemma 2.1
• proof
• Remark 2.4
• Theorem 2.3