### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 343-368 |

Number of pages | 26 |

Journal | Chinese Annals of Mathematics. Series B |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - 2011 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Chinese Annals of Mathematics. Series B*,

*32*(3), 343-368. https://doi.org/10.1007/s11401-011-0647-2

**Invariant measures and asymptotic Gaussian bounds for normal forms of stochastic climate model.** / Yuan, Yuan; Majda, Andrew J.

Research output: Contribution to journal › Article

*Chinese Annals of Mathematics. Series B*, vol. 32, no. 3, pp. 343-368. https://doi.org/10.1007/s11401-011-0647-2

}

TY - JOUR

T1 - Invariant measures and asymptotic Gaussian bounds for normal forms of stochastic climate model

AU - Yuan, Yuan

AU - Majda, Andrew J.

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - Comparison principle

KW - Fokker-Planck equation

KW - Global estimates of probability density function

KW - Invariant measure

KW - Reduced stochastic climate model

UR - http://www.scopus.com/inward/record.url?scp=85027926645&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027926645&partnerID=8YFLogxK

U2 - 10.1007/s11401-011-0647-2

DO - 10.1007/s11401-011-0647-2

M3 - Article

VL - 32

SP - 343

EP - 368

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

SN - 0252-9599

IS - 3

ER -