Elementary treatments of Markov chains, especially those devoted to discrete-time and finite state-space theory, leave the impression that everything is smooth and easy to understand. This exposition of the works of Kolmogorov, Feller, Chung, Kato, and other mathematical luminaries, which focuses on time-continuous chains but is not so far from being elementary itself, reminds us again that the impression is false: an infinite, but denumerable, state-space is where the fun begins. If you have not heard of Blackwell's example (in which all states are instantaneous), do not understand what the minimal process is, or do not know what happens after explosion, dive right in. But beware lest you are enchanted: 'There are more spells than your commonplace magicians ever dreamed of.'

‘Western science is largely based on deducing global properties of systems from the local ones, and differential equations have proved to be a successful mathematical tool for it. It has been realized, however, that even in linear systems there can occur phase transitions that cannot be deduced from the underlying equations. Originally observed in Markov processes, the theory of phase transitions has been recently extended to general master equations. This monograph, building upon Feller's concept of the process boundary and linking it in a novel way with functional analytic tools, provides a refined analysis of the evolution beyond the phase transition. It offers a unique blend of probability and functional analysis addressing a topical problem of nonlocal and emerging properties of infinite-dimensional linear systems and largely extending the existing results.'

Jacek Banasiak - University of Pretoria

‘When I learned about the author's project of a book on Markov chains with denumerable state-space, I was a bit surprised and had serious reservations about it. It seemed to me that in such a simple setting all you can prove is well-known to students who took a first course in Markov processes. But with each page I read I was more convinced that I was thoroughly wrong. … The exposition is clear and reader-friendly. The book requires only a few prerequisites. Moreover, it is autonomous and can be read without extensive knowledge of semigroup theory or stochastic processes. Personally, I hold in high regard self-contained textbooks I can read without consulting other sources. This impressive book belongs to this category.'

Tomasz Szarek - University of Gdansk