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.
A co-publication of the AMS and CBMS.
Readership: Graduate students and research mathematicians interested in stochastic PDEs.
Author(s): Davar Khoshnevisan
Series: CBMS Regional Conference Series in Mathematics Vol. 119
Publisher: American Mathematical Society
Year: 2014
Language: English
Pages: C, viii, 116, B
Tags: Математика;Дифференциальные уравнения;
Chapter 1. Prelude 1
Chapter 2. Wiener integrals 9
2.1. White noise 9
2.2. Stochastic convolutions 11
2.3. Brownian sheet 12
2.4. Fractional Brownian motion 15
Chapter 3. A linear heat equation 19
3.1. A non-random heat equation 19
3.2. The mild solution 22
3.3. Structure theory 22
3.4. Approximation by interacting Brownian particles 28
3.5. Two or more dimensions 30
3.6. Non-linear equations 30
Chapter 4. Walsh–Dalang integrals 33
4.1. The Brownian filtration 33
4.2. The stochastic integral 34
4.3. Integrable random fields 37
Chapter 5. A non-linear heat equation 39
5.1. Stochastic convolutions 40
5.2. Existence and uniqueness of a mild solution 44
5.3. Mild implies weak 50
Chapter 6. Intermezzo: A parabolic Anderson model 53
6.1. Brownian local times 53
6.2. A moment bound 56
Chapter 7. Intermittency 63
7.1. Some motivation 63
7.2. Intermittency and the stochastic heat equation 66
7.3. Renewal theory 67
7.4. Proof of Theorem 7.8 69
Chapter 8. Intermittency fronts 71
8.1. The problem 71
8.2. Some proofs 72
Chapter 9. Intermittency islands 79
9.1. The existence and size of tall islands 79
9.2. A tail estimate 80
9.3. On the upper bound of Theorem 9.1 82
9.4. On the lower bound of Theorem 9.1 83
Chapter 10. Correlation length 87
10.1. An estimate for the length of intermittency islands 87
10.2. A coupling for independence 89
Appendix A. Some special integrals 95
Appendix B. A Burkholder–Davis–Gundy inequality 97
Appendix C. Regularity theory 103
C.1. Garsia’s theorem 103
C.2. Kolmogorov’s continuity theorem 106
Bibliography 111
