Chinese Annals of Mathematics, Series B

, Volume 32, Issue 3, pp 343–368 | Cite as

Invariant measures and asymptotic Gaussian bounds for normal forms of stochastic climate model

  • Yuan YuanEmail author
  • Andrew J. Majda


The systematic development of reduced low-dimensional stochastic climate models from observations or comprehensive high dimensional climate models is an important topic for atmospheric low-frequency variability, climate sensitivity, and improved extended range forecasting. Recently, techniques from applied mathematics have been utilized to systematically derive normal forms for reduced stochastic climate models for low-frequency variables. It was shown that dyad and multiplicative triad interactions combine with the climatological linear operator interactions to produce a normal form with both strong nonlinear cubic dissipation and Correlated Additive and Multiplicative (CAM) stochastic noise. The probability distribution functions (PDFs) of low frequency climate variables exhibit small but significant departure from Gaussianity but have asymptotic tails which decay at most like a Gaussian. Here, rigorous upper bounds with Gaussian decay are proved for the invariant measure of general normal form stochastic models. Asymptotic Gaussian lower bounds are also established under suitable hypotheses.


Reduced stochastic climate model Invariant measure Fokker-Planck equation Comparison principle Global estimates of probability density function 

2000 MR Subject Classification

60H10 60H30 60E99 35Q84 


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Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Department of Mathematics, and Center for Atmospheric Ocean Sciences, Courant InstituteNew York UniversityNew YorkUSA

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