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Discovering Evolution Equations with Applications
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Table of Contents

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 ( ;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 Ito 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 Ito 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 Ito 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.

About the Author

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.

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