Advertisement

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
Article

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.

Keywords

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Berner, J. and Branstator, G., Linear and nonlinear signatures in planetary wave dynamics of an AGCM: probability density functions, J. Atmos. Sci., 64, 2007, 117–136.CrossRefGoogle Scholar
  2. [2]
    Bogachev, V. I., Krylov, N. V. and Rockner, M., Elliptic equations for measures: regularity and global bounds of densities, J. Math. Pures Appl., 85(6), 2006, 743–757.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Bogachev, V. I., Rockner, M. and Shaposhnikov, S. V., On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Part. Diff. Eqs., 26(1112), 2001, 2037–2080.zbMATHCrossRefGoogle Scholar
  4. [4]
    Bogachev, V. I., Rockner, M. and Shaposhnikov, S. V., Lower estimates of densities of solutions of elliptic equations for measures, Doklady Mathematics, 426(2), 2009, 156–161.MathSciNetGoogle Scholar
  5. [5]
    Browder, F. E., Regularity theorems for solutions of partial differential equations with variable coefficients, Proc. Natl. Acad. Sci. USA, 43(2), 1957, 234–236.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Cerrai, S., Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach, Lecture Notes in Mathematics, 1762, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
  7. [7]
    Doob, J. L., Asymptotic properties of markov transition probabilities, Tran. Amer. Math. Soc., 3, 1948, 393–421.MathSciNetGoogle Scholar
  8. [8]
    Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics, 19, A. M. S., Providence, RI, 1998.Google Scholar
  9. [9]
    Fornaro, S., Fusco, N., Metafune, G. and Pallara, D., Sharp upper bounds for the density of some invariant measures, Proc. Roy. Soc. Edinburgh, Sect. A Math., 139, 2009, 1145–1161.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Franzke, C., Majda A. J. and Branstator, G., The origin of nonlinear signatures of planetary wave dynamics: mean phase space tendencies and their information, J. Atmos. Sci., 64(11), 2007, 3987–4003.CrossRefGoogle Scholar
  11. [11]
    Gritsun, A. and Branstator, G., Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem, J. Atmos. Sci., 64, 2007, 2558–2575.CrossRefGoogle Scholar
  12. [12]
    Gritsun, A., Branstator, G. and Majda, A. J., Climate response of linear and quadratic functionals using the fluctuation-dissipation theorem, J. Atmos. Sci., 65, 2008, 2824–2841.CrossRefGoogle Scholar
  13. [13]
    Hormander, L., Hypoelliptic second order differential equations, Acta Math., 119, 1967, 147–171.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Khas’minskii, R. Z., Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations, Theory of Prob. and Its Appl., 9, 1960, 179–196.CrossRefGoogle Scholar
  15. [15]
    Krylov, N. V., Introduction to the Theory of Diffusion Processes, Translations of Mathematical Monographs, 142, A. M. S., Providence, RI, 1995Google Scholar
  16. [16]
    Majda, A. J., Abramov R. and Gershgorin, B., High skill in low frequency climate response through fluctuation dissipation theorems despite structural instability, PNAS, 107(2), 2010, 581–586.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Majda, A. J., Abramov, R. and Grote, M., Information Theory and Stochastics for Multiscale Nonlinear Systems, CRM Monograph Series, 25, A. M. S., Providence, RI, 2005.zbMATHGoogle Scholar
  18. [18]
    Majda, A. J., Franzke, C. and Crommelin, D., Normal forms for reduced stochastic climate models, PNAS, 16(10), 2009, 3649–3653.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Majda, A. J., Franzke C., Fischer A. and Crommelin, D., Distinct metastable atmospheric regimes despite nearly Gaussian statistics: A paradigm model, PNAS, 103(22), 2006, 8309–8314.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Majda, A. J., Franzke, C. and Khouider, B., An applied mathematics perspective on stochastic modelling for climate, Phil. Trans. R Soc. A, 366, 2008, 2429–2455.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Majda, A. J., Gershgorin B. and Yuan, Y., Low frequency response and fluctuation-dissipation theorems: theory and practice, J. Atmos. Sci., 2010, in press. DOI: 10.1175/2009JAS3264.1Google Scholar
  22. [22]
    Majda, A. J., Timofeyev, I. and Vanden-Eijnden, E., A mathematical framework for stochastic climate models, Commun. Pure Appl. Math., 54, 2001, 891–974.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Majda, A. J., Timofeyev, I. and Vanden-Eijnden, E., Models for stochastic climate prediction, PNAS, 96(26), 1999, 14687–14691.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Majda, A. J. and Wang, X. M., Linear response theory for statistical ensembles in complex systems with time-periodic forcing, Comm. Math. Sci., 8(1), 2010, 142–172.Google Scholar
  25. [25]
    Metafune, G., Pallara, D. and Rhandi, A., Global regularity of invariant measures, J. Funct. Anal., 223, 2005, 396–424.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Prato, G. D. and Goldys, B., Elliptic operators on Rd with unbounded coefficients, J. Diff. Eqs., 172(2), 2001, 333–358.CrossRefGoogle Scholar
  27. [27]
    Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  28. [28]
    Sardeshmukh, P. D. and Sura, P., Reconciling non-Gaussian climate statistics with linear dynamics, J. Climate, 22(5), 2009, 1193–1207.CrossRefGoogle Scholar
  29. [29]
    Stephenson, D. B., Hannachi, A. and Oneill, A., On the existence of multiple climate regimes, Q. J. R. Meteorol. Soc., 130, 2004, 583–605.CrossRefGoogle Scholar
  30. [30]
    Sura, P. and Sardeshmukh, P., A global view of non-Gaussian SST variability, J. Phys. Oceonogr., 38, 2008, 639–647.CrossRefGoogle Scholar

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

Personalised recommendations