Invariant Measures and Asymptotic Gaussian Bounds for Normal Forms of Stochastic Climate Model

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 has been 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 departures from Gaussianity but have asymptotic Manuscript received ∗Department of Mathematics, Courant Institute, New York University, 251 Mercer st., New York, NY 10012. E-mail:[email protected] ∗∗Department of Mathematics, and Center for Atmospheric Ocean Sciences, Courant Institute, New York University, 251 Mercer st., New York, NY 10012. E-mail: [email protected]. 2 Y. Yuan and A. J. Majda 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 hypothesis.