UID:
almafu_9960118465602883
Umfang:
1 online resource (x, 223 pages) :
,
digital, PDF file(s).
Ausgabe:
1st ed.
ISBN:
1-108-62352-2
,
1-108-66965-4
,
1-108-67264-7
Serie:
London Mathematical Society student texts ; 93
Inhalt:
The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille-Yosida and Lumer-Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller-Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann-Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.
Anmerkung:
Title from publisher's bibliographic system (viewed on 26 Jul 2019).
,
Cover -- Series page -- Title page -- Copyright page -- Dedication -- Epigraph -- Contents -- Introduction -- 1 Semigroups and Generators -- 1.1 Motivation from Partial Differential Equations -- 1.2 Definition of a Semigroup and Examples -- 1.3 Unbounded Operators and Generators -- 1.3.1 Unbounded Operators and Density of Generators -- 1.3.2 Differential Equations in Banach Space -- 1.3.3 Generators as Closed Operators -- 1.3.4 Closures and Cores -- 1.4 Norm-Continuous Semigroups -- 1.5 The Resolvent of a Semigroup -- 1.5.1 The Resolvent of a Closed Operator -- 1.5.2 Properties of the Resolvent of a Semigroup -- 1.6 Exercises for Chapter 1 -- 2 The Generation of Semigroups -- 2.1 Yosida Approximants -- 2.2 Classifying Generators -- 2.3 Applications to Parabolic PDEs -- 2.3.1 Bilinear Forms, Weak Solutions and the Lax-Milgram Theorem -- 2.3.2 Energy Estimates and Weak Solutions to the Elliptic Problem -- 2.3.3 Semigroup Solution of the Parabolic Problem -- 2.4 Exercises for Chapter 2 -- 3 Convolution Semigroups of Measures -- 3.1 Heat Kernels, Poisson Kernels, Processes and Fourier Transforms -- 3.1.1 The Gauss-Weierstrass Function and the Heat Equation -- 3.1.2 Brownian Motion and Itô's Formula -- 3.1.3 The Cauchy Distribution, the Poisson Kernel and Laplace's Equation -- 3.2 Convolution of Measures and Weak Convergence -- 3.2.1 Convolution of Measures -- 3.2.2 Weak Convergence -- 3.3 Convolution Semigroups of Probability Measures -- 3.4 The Lévy-Khintchine Formula -- 3.4.1 Stable Semigroups -- 3.4.2 Lévy Processes -- 3.5 Generators of Convolution Semigroups -- 3.5.1 Lévy Generators as Pseudo-Differential Operators -- 3.6 Extension to L[sup(p)] -- 3.7 Exercises for Chapter 3 -- 4 Self-Adjoint Semigroups and Unitary Groups -- 4.1 Adjoint Semigroups and Self-Adjointness -- 4.1.1 Positive Self-Adjoint Operators.
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4.1.2 Adjoints of Semigroups on Banach Spaces -- 4.2 Self-Adjointness and Convolution Semigroups -- 4.3 Unitary Groups, Stone's Theorem -- 4.4 Quantum Dynamical Semigroups -- 4.5 Exercises for Chapter 4 -- 5 Compact and Trace Class Semigroups -- 5.1 Compact Semigroups -- 5.2 Trace Class Semigroups -- 5.2.1 Hilbert-Schmidt and Trace Class Operators -- 5.2.2 Trace Class Semigroups -- 5.2.3 Convolution Semigroups on the Circle -- 5.2.4 Quantum Theory Revisited -- 5.3 Exercises for Chapter 5 -- 6 Perturbation Theory -- 6.1 Relatively Bounded and Bounded Perturbations -- 6.1.1 Contraction Semigroups -- 6.1.2 Analytic Semigroups -- 6.2 The Lie-Kato-Trotter Product Formula -- 6.3 The Feynman-Kac Formula -- 6.3.1 The Feynman-Kac Formula via the Lie-Kato-Trotter Product Formula -- 6.3.2 The Feynman-Kac Formula via Itô's Formula -- 6.4 Exercises for Chapter 6 -- 7 Markov and Feller Semigroups -- 7.1 Definitions of Markov and Feller Semigroups -- 7.2 The Positive Maximum Principle -- 7.2.1 The Positive Maximum Principle and the Hille-Yosida-Ray Theorem -- 7.2.2 Crash Course on Distributions -- 7.2.3 The Courrège Theorem -- 7.3 The Martingale Problem -- 7.3.1 Sub-Feller Semigroups -- 8 Semigroups and Dynamics -- 8.1 Invariant Measures and Entropy -- 8.1.1 Invariant Measures -- 8.1.2 Entropy -- 8.2 Semidynamical Systems -- 8.2.1 Koopmanism -- 8.2.2 Dynamical Systems and Differential Equations -- 8.3 Approaches to Irreversibility -- 8.3.1 Dilations of Semigroups -- 8.3.2 The Misra-Prigogine-Courbage Approach -- 9 Varopoulos Semigroups -- 9.1 Prelude - Fractional Calculus -- 9.2 The Hardy-Littlewood-Sobolev Inequality and Riesz Potential Operators -- 9.3 Varopoulos Semigroups -- 9.4 Varopoulos's Theorem -- 9.5 Nash Inequality, Symmetric Diffusion Semigroups and Heat Kernels -- Notes and Further Reading -- Appendix A The Space C[sub(0)](R[sup(d)]).
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Appendix B The Fourier Transform -- Appendix C Sobolev Spaces -- Appendix D Probability Measures and Kolmogorov's Theorem on Construction of Stochastic Processes -- Appendix E Absolute Continuity, Conditional Expectation and Martingales -- E.1 The Radon-Nikodym Theorem -- E.2 Conditional Expectation -- E.3 Martingales -- Appendix F Stochastic Integration and Itô's Formula -- F.1 Stochastic Integrals -- F.2 Itô's Formula -- Appendix G Measures on Locally Compact Spaces - Some Brief Remarks -- References -- Index.
Weitere Ausg.:
ISBN 1-108-48309-7
Sprache:
Englisch
URL:
https://doi.org/10.1017/9781108672641
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