Stochastic Partial Differential Equations and Environmental and Geophysical Modeling

Course Description

In this lecture series the speaker will discuss stochastic partial differential equations where the solution is subjected to certain additional constraints. Examples will include fluid dynamics on spherical surfaces modeling ocean and atmospheric dynamics. Suitable stochastic integration theory will also be developed along with study of dynamical systems aspects such as random attractors and invariant measures.

Topics:

• Derivation of the stochastic Navier-Stokes Equations and their Geophysical applications, e.g. oceanography
  
  
• Stochastic Ito integrals: definition and basic properties; Burkholder inequality for Vector valued integrals and applications
  
  
• Existence results for stochastic Navier-Stokes Equations
  
  
• Existence results for stochastic partial differential equations with nonlinear constraints
  
  
• Existence and properties of; random attractors, invariant measures, etc.

Course Description

White noise theory was originally developed by T.Hida for the case of Brownian motion. It may be regarded as a stochastic distribution theory which, combined with the Wick product, extends the classical Itô calculus, both to the anticipating case (Skorohod integrals) and to the multi-parameter case (random fields). Like the classical distribution theory of Laurent Schwartz, which is useful in the study of deterministic PDEs, the white noise theory is useful in the study of SPDEs.

Recently there has been an increased interest in stochastic models driven by Lévy processes (i.e., processes with stationary independent increments), One reason for this is that in applications such models can be made more realistic than the classical Brownian motion based models. This makes it natural to request a white noise theory for such processes as well.

Topics:

• In the first part of these lectures we start by giving an introduction to the classical white noise theory based on Brownian motion, and then we proceed to develop a similar theory for a general Lévy process. Then we show how this theory can be applied to solve SPDEs driven by Brownian motion and/or general Lévy processes.
  
  
• In the second part of the lectures we introduce the Malliavin calculus in a white noise context, again first for Brownian motion and then for a general Lévy process, and we give some applications to mathematical finance.

Course Description

The course consists of ten hour-lectures on the analysis of stochastic hyperbolic equations and their applications to wave propagation through random or turbulent media. The topics to be covered may include the following subjects:

• Continuous random fields, stochastic integral and semi-martingales with spatial parameters.
  
  
• First-order stochastic hyperbolic systems, energy inequalities and existence of path-wise solutions, method of characteristics and stochastic flows of diffeomorphisms, stochastic ray equations and a limit theorem.
  
  
• Linear wave equations with random coefficients, weak solution as a generalized Brownian functional, energy inequalities, existence and regularity of strong solutions, moment equations and propagation of mutual coherence functions.
  
  
• Semilinear stochastic wave equations, energy estimates, local and global solutions, the large-time asymptotic behavior of solutions: boundedness, stabilities and the existence of invariant measures.
  
  
• Selected applications to atmospheric and oceanic wave propagation problems arising from geophysical models.

Course Description

An important problem of fluid dynamics, almost open, is concerned with the foundations of the Statistical Theory of Turbulence, not to say the knowledge of the exact, not only approximate, laws of it. The role of vortex dynamics in turbulence is well established in fluid mechanics. This course will introduce modern statistical/stochastic analysis of vortex dynamics and vortex filaments.

Topics:

• The first part of the course will be devoted to a summary of some statistical law, like K41 and multi-fractal scaling, and a discussion of the problem of the connection between the (stochastic) Navier-Stokes equations and these laws.
  
  
• The second part of the course aims at the construction of a rigorous ensemble of velocity fields with two properties: its realizations are made of geometrical structures, typically of vortex filament type, similar to those observed in numerical simulations; its statistics reproduce statistical laws of turbulence.
  
  
• Several open problems will be discussed, including those of vortex dynamics and the possible relations with the invariant measures of stochastic Navier-Stokes equations.

Course Description

This course will deal with the foundations of stochastic partial differential equations. Linear and nonlinear partial differential equations subjected to Gaussian and Poisson type noise will be studied. Applications and motivating examples will come from control and filtering theory.

Topics: