Analysis of Stochastic Partial Differential Equations (CBMS Regional Conference Series in Mathematics) (CBMS Regional Conference Series in Mathematics, 119)

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. Khoshnevisan, D. Analysis of stochastic partial differential equations, CBMS 119 (AMS,2014) ......Page 3 Copyright ......Page 4 Contents ......Page 6 Chapter 1. Prelude 1 ......Page 8 2.1. White noise 9 ......Page 16 2.2. Stochastic convolutions 11 ......Page 18 2.3. Brownian sheet 12 ......Page 19 2.4. Fractional Brownian motion 15 ......Page 22 3.1. A non-random heat equation 19 ......Page 26 3.3. Structure theory 22 ......Page 29 3.4. Approximation by interacting Brownian particles 28 ......Page 35 3.6. Non-linear equations 30 ......Page 37 4.1. The Brownian filtration 33 ......Page 40 4.2. The stochastic integral 34 ......Page 41 4.3. Integrable random fields 37 ......Page 44 Chapter 5. A non-linear heat equation 39 ......Page 46 5.1. Stochastic convolutions 40 ......Page 47 5.2. Existence and uniqueness of a mild solution 44 ......Page 51 5.3. Mild implies weak 50 ......Page 57 6.1. Brownian local times 53 ......Page 60 6.2. A moment bound 56 ......Page 63 7.1. Some motivation 63 ......Page 70 7.2. Intermittency and the stochastic heat equation 66 ......Page 73 7.3. Renewal theory 67 ......Page 74 7.4. Proof of Theorem 7.8 69 ......Page 76 8.1. The problem 71 ......Page 78 8.2. Some proofs 72 ......Page 79 9.1. The existence and size of tall islands 79 ......Page 86 9.2. A tail estimate 80 ......Page 87 9.3. On the upper bound of Theorem 9.1 82 ......Page 89 9.4. On the lower bound of Theorem 9.1 83 ......Page 90 10.1. An estimate for the length of intermittency islands 87 ......Page 94 10.2. A coupling for independence 89 ......Page 96 Appendix A. Some special integrals 95 ......Page 102 Appendix B. A Burkholder-Davis-Gundy inequality 97 ......Page 104 C.l. Garsia’s theorem 103 ......Page 110 C.2. Kolmogorov’s continuity theorem 106 ......Page 113 Bibliography 111 ......Page 118 cover......Page 1 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 a la Norbert Wiener, an infinite-dimensional Ito-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. Prelude -- Wiener Integrals -- A Linear Heat Equation -- Walsh-damang Integrals -- A Non-linear Heat Equation -- Intermezzo: A Parabolic Anderson Model -- Intermittency -- Intermittency Fronts -- Intermittency Islands -- Correlation Length -- Appendix A: Some Special Integrals -- Appendix B: A Burkholder-davis-gundy Inequality -- Appendix C: Reguarity Theory. Davar Khoshnevisan. Nsf-cbms Regional Conference In The Mathematical Sciences: Analysis Of Stochastic Partial Differential Equations Held At Michigan State University, East Lansing, Michigan, August 19-23, 2013--verso Of Title Page. Includes Bibliographical References.