Bobrowski A. Generators of Markov Chains: From a Walk in the Interior to a Dance on the Boundary

Cambridge University Press, 2021. — 280 p. — (Cambridge Studies in Advanced Mathematics 190). — ISBN 978-1-108-49579-0.The theory of Markov chains, whether time discrete or time continuous, is one of the integral parts of the theory of stochastic processes. This book, however, is not devoted to the popular part of this rich theory, so the reader will not learn here about recurrent and transient states, ergodic theorems, or convergence to equilibrium. (In Arsheesh’s words, thus, we will ask questions that do not concern us, having no business in mind, in the hope that we will somehow reach Narnia and the North.) Instead, we focus on the equally intriguing question of how a continuous-time Markov chain may be described by means of its Kolmogorov (intensity) matrix or its generator, and we study the interplay between the notions just named. We argue in particular that, despite their popularity, Kolmogorov (intensity) matrices are less suitable for such description than generators. Whereas, in their relative simplicity, they allow an intuitive formulation of processes, in general, they fail to describe more delicate phenomena.Nontechnical Introduction
A Guided Tour through the Land of Operator Semigroups
Generators versus Intensity Matrices
Boundary Theory: Core Results
Boundary Theory Continued
The Dual Perspective