During June 13 - July 1, 2005 the University of Wyoming, Laramie will be the host of the Institute for Mathematics and its Applications (IMA) summer graduate program in mathematics. The course will concentrate on Stochastic Partial Differential Equations and Environmental and Geophysical Modeling.
This program is open to graduate students from IMA Participating Institutions. Students are nominated by their department head. Participating institution department heads nominate graduate students from their institution by an e-mail to firstname.lastname@example.org with the students' names and e-mail addresses.
Those students may then register by filling out the registration form. Places are guaranteed for two graduate students from each participating institution, with additional students accommodated as space allows.
Stochastic Navier Stokes Equations on Riemannian Manifolds and Stochastic Partial Differential Equations with Nonlinear Constraints: Zdzislaw Brzezniak, University of Hull
Length: three weeks, June 13 - July 1
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.
- 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.
White Noise Theory and Malliavin Calculus for Lévy Processes and Applications to SPDEs and Finance: Bernt Øksendal, University of Oslo Norway
Length: two weeks, June 20-July 1
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.
- 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.
Stochastic Hyperbolic Equations and Random Wave Propagation: Pao-Liu Chow, Wayne State University Detroit
Length: two weeks, June 13-24
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.
Statistical Theory of Turbulence and Modeling of Vortex Dynamics: Franco Flandoli, University of Dini
Length: two weeks, June 20-July 1
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.
- 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.
Stochastic Partial Differential Equations: Theory & Applications: Jerzy Zabczyk, Polish Academy of Sciences
Length: two weeks, June 13-24
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.
- The first two lectures will be on stochastic ordinary equations (with Gaussian noise) their invariant measures and on examples (Ornstein-Uhlenbeck processes, equations with multiplicative noise etc.).
- The third and the fourth lecture would be on linear stochastic PDEs including heat and wave equations, their invariant measures and, at the end, on abstract Cauchy problem in its weak and integral forms.
- Nonlinear heat and wave equations with various noises will be the content of the fifth and of the sixth lecture.
- The next two days will be on stochastic PDEs with impulsive noises. This will require an introduction to the stochastic integration with respect to Poissonian random measures.
- Two final lectures will be devoted to filtering and control of infinite dimensional systems.