We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.
- Nonlinear and Stochastic Reduction: Markovian and non-Markovinan stochastic closures. Parameterizing manifolds and small-scale parameterizations in spatially extended systems. Ruelle-Pollicott resonances and reduced mixing spectrum.
- Random Dynamical Systems: Stochastic invariant manifolds and their numerical approximations. Random attractors and time-dependent invariant measures arising in dissipative stochastic systems.
- Data-Driven Stochastic Modeling: Data-adaptive harmonic spectra and multilayer Stuart-Landau models. Applications to Arctic sea ice modeling, and wind-driven ocean gyres.
- Optimal Control of Evolution Equations: Reduced-order models for the optimal control of PDEs and delay differential equations: analysis and computation.
- Geophysical Fluid Dynamics and Climate Dynamics: Natural climate variability. Low-order stochastic modelling of the El Niño-Southern Oscillation. Prediction of stochastic systems. Application of the geometric and ergodic theory of dynamical systems to climate dynamics.
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