American Institute of Mathematical Sciences

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October  2010, 26(4): 1241-1268. doi: 10.3934/dcds.2010.26.1241

Singular perturbation systems with stochastic forcing and the renormalization group method

Nathan Glatt-Holtz 1, and  Mohammed Ziane 2,

1. 

Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, United States
2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

Received  November 2008 Revised  April 2009 Published  December 2009

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.
Keywords: Stochastic Analysis, Renormalization Group Method, White Noise, Singular Perturbation..
Mathematics Subject Classification: 60H1.
Citation: Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
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