Author:
Applebaum, David, 1956- author.
ImprintCambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019.
Descriptionx, 223 pages ; 23 cm.
Note:Semigroups and generators -- The generation of semigroups -- Convolution semigroups of measures -- Self adjoint semigroups and unitary groups -- Compact and trace class semigroups -- Perturbation theory -- Markov and Feller semigroups -- Semigroups and dynamics -- Varopoulos semigroups -- Appendices: A. The space C0(Rd) -- B. The Fourier transform -- C. Sobolev spaces -- D. Probability measures and Kolmogorov's theorem on construction of stochastic processes -- E. Absolute continuity, conditional expectation and martingales -- F. Stochastic integration and Itô's formula -- G. Measures on locally compact spaces: some brief remarks.
Bibliography Note:Includes bibliographical references (pages 214-218) and index.
Note: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 Le?vy 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.
