Abstract
This article examines a class of singular perturbation systems in the presence of a small white noise. Modifying the renormalization group procedure developed by Chen, Goldenfeld and Oono [6], we derive an associated reduced system which we use to construct an approximate solution that separates scales. Rigorous results demonstrating that these approximate solutions remain valid with high probability on large time scales are established. As a special case we infer new small noise asymptotic results for a class of processes exhibiting a physically motivated cancellation property in the nonlinear term. These results are applied to some concrete perturbation systems arising in geophysical fluid dynamics and in the study of turbulence. For each system we exhibit the associated renormalization group equation which helps decouple the interactions between the different scales inherent in the original system.
Highlights
Perturbation theory has long played an important role in applied analysis
A basic challenge in the study of perturbed dynamical systems is that the unperturbed problem may exhibit fundamentally different quantitative and qualitative behaviors, for large time scales or near certain boundaries
As we shall see below, when the noise is “small” (i.e. when m > 0 in the original system (5) or (8)) a approximate solution without stochastic terms can be shown to be valid on the time scale 1/ǫ
Summary
Perturbation theory has long played an important role in applied analysis. Perturbed systems can provide the relevant setting for the study of physical phenomena exhibiting multiple spatial and temporal scales. Renormalization Group Method, White Noise, Stochastic Analysis, Singular Perturbation. One interesting consequence of these results is that when A ≡ 0 in the original system (1), (4) can be interpreted as a small noise asymptotic result While such results are classical for systems with Lipschitz nonlinear terms (see [11] or [8]) our result allows us to address the physically important case when one can only expect cancellations in the nonlinear portion of the equation. As we shall see below, when the noise is “small” (i.e. when m > 0 in the original system (5) or (8)) a approximate solution without stochastic terms can be shown to be valid on the time scale 1/ǫ. The analysis must consider the antisymmetric and positive semidefinite cases separately
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