1st Edition
Discovering Evolution Equations with Applications Volume 2-Stochastic Equations
Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations.
The text begins with hands-on introductions to the essentials of real and stochastic analysis. It then develops the theory for homogenous one-dimensional stochastic ordinary differential equations (ODEs) and extends the theory to systems of homogenous linear stochastic ODEs. The next several chapters focus on abstract homogenous linear, nonhomogenous linear, and semi-linear stochastic evolution equations. The author also addresses the case in which the forcing term is a functional before explaining Sobolev-type stochastic evolution equations. The last chapter discusses several topics of active research.
Each chapter starts with examples of various models. The author points out the similarities of the models, develops the theory involved, and then revisits the examples to reinforce the theoretical ideas in a concrete setting. He incorporates a substantial collection of questions and exercises throughout the text and provides two layers of hints for selected exercises at the end of each chapter.
Suitable for readers unfamiliar with analysis even at the undergraduate level, this book offers an engaging and accessible account of core theoretical results of stochastic evolution equations in a way that gradually builds readers’ intuition.
A Basic Analysis Toolbox
Some Basic Mathematical Shorthand
Set Algebra
Functions
The Space (R, |·|)
Sequences in (R, |·|)
The Spaces (RN, ||·||RN) and (MN(R), ||·||MN(R))
Abstract Spaces
Elementary Calculus in Abstract Spaces
Some Elementary ODEs
A Handful of Integral Inequalities
Fixed-Point Theory
The Bare-Bone Essentials of Probability Theory
Formalizing Randomness
R-Valued Random Variables
Introducing the Space L2 (Ω;R)
RN-Valued Random Variables
Conditional Probability and Independence
Conditional Expectation—A Very Quick Description
Stochastic Processes
Martingales
The Wiener Process
Summary of Standing Assumptions
Looking Ahead
Linear Homogenous Stochastic Evolution Equations in R
Random Homogenous Stochastic Differential Equations
Introducing the Lebesgue and Itó Integrals
The Cauchy Problem—Formulation
Existence and Uniqueness of a Strong Solution
Continuous Dependence on Initial Data
Statistical Properties of the Strong Solution
Some Convergence Results
A Brief Look at Stability
A Classical Example
Looking Ahead
Homogenous Linear Stochastic Evolution Equations in RN
Motivation by Models
Deterministic Linear Evolution Equations in RN
Exploring Two Models
The Lebesgue and Itó Integrals in RN
The Cauchy Problem—Formulation
Existence and Uniqueness of a Strong Solution
Continuous Dependence on Initial Data
Statistical Properties of the Strong Solution
Some Convergence Results
Looking Ahead
Abstract Homogenous Linear Stochastic Evolution Equations
Linear Operators
Linear Semigroup Theory—Some Highlights
Probability Theory in the Hilbert Space Setting
Random Homogenous Linear SPDEs
Bochner and Itó Integrals
The Cauchy Problem—Formulation
The Basic Theory
Looking Ahead
Nonhomogenous Linear Stochastic Evolution Equations
Finite-Dimensional Setting
Nonhomogenous Linear SDEs in R
Nonhomogenous Linear SDEs in RN
Abstract Nonhomogenous Linear SEEs
Introducing Some New Models
Looking Ahead
Semi-Linear Stochastic Evolution Equations
Motivation by Models
Some Essential Preliminary Considerations
Growth Conditions
The Cauchy Problem
Models Revisited
Theory for Non-Lipschitz-Type Forcing Terms
Looking Ahead
Functional Stochastic Evolution Equations
Motivation by Models
Functionals
The Cauchy Problem
Models—New and Old
Looking Ahead
Sobolev-Type Stochastic Evolution Equations
Motivation by Models
The Abstract Framework
Semi-Linear Sobolev Stochastic Equations
Functional Sobolev SEEs
Beyond Volume 2
Fully Nonlinear SEEs
Time-Dependent SEEs
Quasi-Linear SEEs
McKean-Vlasov SEEs
Even More Classes of SEEs
Bibliography
Index
Guidance for Selected Exercises appears at the end of each chapter.
Biography
Mark A. McKibben is a professor of mathematics and computer science at Goucher College. He serves as a referee for more than 30 journals and has published numerous articles in peer-reviewed journals. Dr. McKibben earned a Ph.D. in mathematics from Ohio University. His research interests include nonlinear and stochastic evolution equations.
