Prepared by Roger Temam (Indiana University) and Nathan Glatt-Holtz (University of Minnesota/Virginia Tech)

Last week a workshop was held at the American Institute of Mathematics (AIM) in Palo Alto, California, around the theme of stochastic PDEs and applications in climate and weather modeling:

**“Stochastic in Geophysical Fluid Dynamics: Mathematical foundations and physical underpinnings.”**

The workshop brought together a lively mix of specialists in climate modeling and weather prediction alongside experts in the fields of deterministic and stochastic partial differential equations.

Stochastic Differential Equations (SDEs) have their origins in the study of Brownian motion the irregular motion of particles under the influence of random bombardment first observed by the botanist Robert Brown in the 19th century and later revisited by Albert Einstein in 1904.

The mathematical theory traces its origins to the work of Norbert Wiener and Andrey Kolmogorov in the 1920’s. The key discovery of the stochastic integral and an associated ‘stochastic calculus’ which lies at the center of the theory occurred in the early 1940’s independently by Kiyoshi Itō and by Wolfgang Doeblin.

Roughly speaking Browian Motion {B(t)}_{t \geq 0} is a time evolving random process which evolves according the the normal (Gaussian) distribution with mean zero and variance t. A key property is the Browian motion has ‘independent increments’ that is B(t) – B(s)is independent of B(s) – B( r) for any t> s> r. This last property gives Brownian motion it’s irregular character.

Stochastic differential equations are equations driven by the derivative of brownian motion, ‘white noise’ which is used in modeling often as a proxy for uncertainty. Indeed today SDEs are widely used in diverse modeling applications ranging from biological population models to hedging in finance, where they provide the foundation for the infamous Black–Scholes–Merton Model. From a theoretical point of view they play a central role in the modern theory of probability and stochastic processes.

The study of Stochastic Partial Differential equations (SPDEs) traces its origins to the 1960. They first appeared in the theory of filtering (the optimal melding of empirical data with a dynamical model to more accurately predict the state of a physical system), in the study of turbulence in fluid flows and in biological models of neurons.

Since the 1970 SDEs and SPDEs have played an increasingly large role in climate and weather applications.As it was explained during the wokshop, SDEs and SPDEs are introduced for various reasons: to account for uncertainties in the various data introduced in the codes (e.g. initial and boundary conditions); they are also used also to account for the uncertainties in the physics for some complex phenomena (such as the parameterization of the clouds, or of the strength of the wind above the oceans); it is also used to account for the errors due to the discretizations. For example the typical mesh used in the General Circulation Models (CGM) is nowadays of about 15 to 25 km; this does not allow for a fine description of the cloudiness in the numerical cells, and statistics is used to give an averaged local description.

The week was highlighted by many interesting lectures which promoted significant dialogue and cross-pollination of ideas between the two communities. These lectures were followed by lengthy moderated discussions which provided an opportunity for both sides to ask `naive’ questions and to formulate novel research problems and directions. This was a rare opportunity for both sides to interact in an informal setting.

In addition to the lectures of Mohammed Ziane and Joe Tribbia (see our previous post) some of the week’s talks included:

1) Cecile Penland (NOAA) gave an overview of the current state of uncertainty quantification in operational weather models and discussed how these estimates could be improved with the use of more sophisticated stochastic methods. Her lecture was a useful reminder to the mathematical community of the daunting complexity of the existing numerical models and uneven quality of atmospheric data available.

2) Boris Rozovskii (Brown) outlined recent developments in the use of wiener chaos expansions in the numerical and theoretical study of stochastic partial differential equations.

3) Antonio Navarra (Centro Euro-Mediterraneo sui Cambiamenti Climatici)-Described recent developments in the use of Feynman path integrals to calculate probability distributions for certain stochastic climate models.

4) David Neelin (UCLA) Discussed approaches to parameter estimation and sensitivity in precipitation models.

5) Mickael Chekroun (UCLA/Hawaii) Introduced theoretical and practical approaches to Markov approximation of chaotic models. He discussed applications in his joint work with David Neelin.

6) Franco Flandoli (Pisa) Overviewed the mathematical foundation of the Kolmogorov equations and discussed practical challenges for the computation of probability distributions for nonlinear Stochastic PDEs.

7) Susan Friedlander (USC) Discussed recent developments in the understanding of the inertial structure of the 3d Navier-Stokes equations and explained connections to turbulent flows. Motivated by this work she also introduced some novel `shell models’ which permit the recovery of the fundamental statistical quantities arising in 3D turbulence theory.

8) Armen Shirikyan (Université de Cergy-Pontoise, Paris)- Discussed ergodic and mixing properties of the stochastic and randomly kick forced 2d Navier-Stotkes equations and related models. He also discussed normal approximation and large deviations for these models.

9) Peter Kloeden (Goethe-Universität)- Discussed theoretical and practical issues involved with the numerical simulation of finite and infinite dimensional stochastic equations using Taylor expansion methods.

10) Nathan Glatt-Holtz (University of Minnesota/Virginia Tech)- Discussed inviscid limits for stochastic fluids equations and relationships with Turbulence theory.