After recalling essentials of analysis — including functional analysis, convexity, distribution theory and interpolation theory — this book handles two topics in detail: Fourier analysis, with emphasis on positivity and also on some function spaces and multiplier theorems; and one-parameter operator semigroups with emphasis on Feller semigroups and Lp-sub-Markovian semigroups. In addition, Dirichlet forms are treated. The book is self-contained and offers new material originated by the author and his students.
Sample Chapter(s)
Introduction: Pseudo Differential Operators and Markov Processes (207 KB)
Chapter 1: Introduction (190 KB)
Contents:
- Essentials from Analysis:
- Calculus Results
- Convexity
- Some Interpolation Theory
- Fourier Analysis and Convolution Semigroups:
- The Paley–Wiener–Schwartz Theorem
- Bounded Borel Measures and Positive Definite Functions
- Convolution Semigroups and Negative Definite Functions
- The Lévy–Khinchin Formula for Continuous Negative Definite Functions
- Bernstein Functions and Subordination of Convolution Semigroups
- Fourier Multiplier Theorems
- One Parameter Semigroups:
- Strongly Continuous Operator Semigroups
- Subordination in the Sense of Bochner for Operator Semigroups
- Generators of Feller Semigroups
- Dirichlet Forms and Generators of Sub-Markovian Semigroups
- and other papers
Readership: Graduate students, researchers and lecturers in analysis & differential equations, stochastics, probability & statistics, and mathematical physics.
