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

Yuan Yuan, Andrew J. Majda

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)343-368
Number of pages26
JournalChinese Annals of Mathematics. Series B
Issue number3
StatePublished - May 2011


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

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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