1st Edition
Discovering Evolution Equations with Applications Volume 2-Stochastic Equations
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.
