Analysis of Stochastic Partial Differential Equations
About this Title
Davar Khoshnevisan, University of Utah, Salt Lake City, UT
Publication: CBMS Regional Conference Series in Mathematics
Publication Year 2014: Volume 119
ISBNs: 978-1-4704-1547-1 (print); 978-1-4704-1712-3 (online)
MathSciNet review: MR3222416
MSC: Primary 60H15; Secondary 35R60, 60H30
The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance.
The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a “random noise,” also known as a “generalized random field.” At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe.
The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals à la Norbert Wiener, an infinite-dimensional Itô-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts.
There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.
Graduate students and research mathematicians interested in stochastic PDEs.
Table of Contents
- 1. Prelude
- 2. Wiener integrals
- 3. A linear heat equation
- 4. Walsh-Dalang integrals
- 5. A non-linear heat equation
- 6. Intermezzo: A parabolic Anderson model
- 7. Intermittency
- 8. Intermittency fronts
- 9. Intermittency islands
- 10. Correlation length
- Appendix A. Some special integrals
- Appendix B. A Burkholder-Davis-Gundy inequality
- Appendix C. Regularity theory