This book combines the spirit of a textbook with that of a monograph on the topic of semigroups and their applications. It is expected to have potential readers across a broad spectrum that includes operator theory, partial differential equations, harmonic analysis, probability and statistics, and classical and quantum mechanics.

A reasonable amount of familiarity with real analysis, including the Lebesgue-integration theory, basic functional analysis, and bounded linear operators is assumed. However, any discourse on a theory of semigroups needs an introduction to unbounded linear operators, some elements of which have been included in the Appendix, along with the basic ideas of the Fourier transform and of Sobolev spaces. Chapters 4–6 contain advanced material not often found in textbooks but which have many interesting applications, such as the Feynman–Kac formula, the central limit theorem, and the construction of Markov semigroups. The exercises are given in the text as the topics are developed so that interested readers can solve these while learning that topic.