Stochastic Partial Differential Equations

Description

Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions.  As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected to the material discussed at a particular place in the text. The questions usually ask to verify something, so that the reader already knows the answer and, if pressed for time, can move on. Accordingly, no solutions are provided, but there are often hints on how to proceed. The book will be of interest to everybody working in the area of stochastic analysis, from beginning graduate students to experts in the field. Preface Contents 1 Introduction 1.1 Getting Started 1.1.1 Conventions and Notations 1.1.2 Dealing with Noise 1.1.3 A Few Useful Equalities 1.1.4 A Few Useful Inequalities 1.2 Some Sources of SPDEs 1.2.1 Biology 1.2.2 Classical Probability Theory 1.2.3 Economics and Finance 1.2.4 Engineering 1.2.5 Physics 1.2.6 Literature 1.2.7 The Structure of This Book 2 Basic Ideas 2.1 Some Useful Facts 2.1.1 Continuity of Random Functions 2.1.2 Connection Between the Itô and Stratonovich Integrals 2.1.3 Random Change of Variables in Random Functions 2.1.4 Problems 2.2 Classification of SPDEs 2.2.1 SPDEs as Stochastic Equations 2.2.2 SPDEs as Partial Differential Equations 2.2.3 Various Notions of a Solution 2.2.4 Problems 2.3 Closed-Form Solutions 2.3.1 Heat Equation 2.3.1.1 Further Directions 2.3.2 Wave Equation 2.3.3 Poisson Equation 2.3.4 Nonlinear Equations 2.3.5 Problems 3 Stochastic Analysis in Infinite Dimensions 3.1 An Overview of Functional Analysis 3.1.1 Spaces 3.1.2 Linear Operators 3.1.3 Problems 3.2 Random Processes and Fields 3.2.1 Fields (No Time Variable) 3.2.2 Processes 3.2.3 Martingales 3.2.4 Problems 3.3 Stochastic Integration 3.3.1 Construction of the Integral 3.3.2 Itô Formula 3.3.3 Problems 4 Linear Equations: Square-Integrable Solutions 4.1 A Summary of SODEs and Deterministic PDEs 4.1.1 Why Square-Integrable Solutions? 4.1.2 Classification of PDEs 4.1.2.1 Second-Order PDEs in Two Variables and Conic Sections 4.1.3 Proving Well-Posedness of Linear PDEs 4.1.4 Well-Posedness of Abstract Equations 4.1.5 Problems 4.2 Stochastic Elliptic Equations 4.2.1 Existence and Uniqueness of Solution 4.2.2 An Example and Further Directions 4.2.3 Problems 4.3 Stochastic Hyperbolic Equations 4.3.1 Existence and Uniqueness of Solution 4.3.2 Further Directions 4.3.3 Problems 4.4 Stochastic Parabolic Equations 4.4.1 Existence and Uniqueness of Solution 4.4.2 A Change of Variables Formula 4.4.3 Probabilistic Representation of the Solution, Part I: Method of Characteristics 4.4.4 Probabilistic Representation of the Solution, Part II: Measure-Valued Solutions and the Filtering Problem 4.4.5 Further Directions 4.4.6 Problems 5 The Polynomial Chaos Method 5.1 Stationary Wiener Chaos 5.1.1 Cameron-Martin Basis 5.1.2 Elementary Operators on Cameron-Martin Basis 5.1.3 Elements of Stationary Malliavin Calculus 5.1.4 Problems 5.2 Stationary SPDEs 5.2.1 Definitions and Basic Examples 5.2.2 Solving Stationary SPDEs by Weighted Wiener Chaos 5.2.2.1 Existence and Uniqueness of Solutions 5.3 Elements of Malliavin Calculus for Brownian Motion 5.3.1 Cameron-Martin Basis for Scalar Brownian Motion 5.3.1.1 Cameron-Martin Basis for Brownian Motion in a Hilbert Space 5.3.2 The Malliavin Derivative and Its Adjoint 5.4 Wiener Chaos Solutions for Parabolic SPDEs 5.4.1 The Propagator 5.4.2 Special Weights and The S-Transform 5.5 Further Properties of the Wiener Chaos Solutions 5.5.1 White Noise and Square-Integrable Solutions 5.5.2 Additional Regularity 5.5.3 Probabilistic Representation 5.6 Examples 5.6.1 Wiener Chaos and Nonlinear Filtering 5.6.2 Passive Scalar in a Gaussian Field 5.6.3 Stochastic Navier-Stokes Equations 5.6.4 First-Order Itô Equations 5.6.5 Problems 5.7 Distribution Free Stochastic Analysis 5.7.1 Distribution Free Polynomial Chaos 5.7.2 Distribution Free Malliavin Calculus 5.7.2.1 The Malliavin Derivative 5.7.3 Adapted Stochastic Processes 5.7.3.1 Itô-Skorokhod Isometry 5.7.4 Stochastic Differential Equations 5.7.4.1 Wick Exponential 5.7.4.2 Linear SDEs 5.7.4.3 Linear Parabolic SPDEs 5.7.4.4 Stationary SPDEs 5.7.4.5 Weighted Norms 5.7.4.6 Wick-Nonlinear SPDEs 6 Parameter Estimation for Diagonal SPDEs 6.1 Examples and General Ideas 6.1.1 An Oceanographic Model and Its Simplifications 6.1.2 Long Time vs Large Space 6.1.3 Problems 6.2 Maximum Likelihood Estimator (MLE):One Unknown Parameter 6.2.1 The Forward Problem 6.2.2 Simplified MLE (sMLE) and MLE 6.2.3 Consistency and Asymptotic Normality of sMLE 6.2.4 Asymptotic Efficiency of the MLE 6.2.5 Problems 6.3 Several Parameters and Further Directions 6.3.1 The Heat Balance Equation 6.3.2 One Parameter: Beyond an SPDE 6.3.3 Several Parameters 6.3.4 Problems Problems: Answers, Hints, Further Discussions Problems of Chap. 2 Problems of Chap. 3 Problems of Chap. 4 Problems of Chap. 5 Problems of Chap. 6 References List of Notations Index