Generators of Markov Chains
From a Walk in the Interior to a Dance on the Boundary

A clear explanation of what an explosive Markov chain does after it passes through all available states in finite time.

Adam Bobrowski (Author)

9781108495790, Cambridge University Press

Hardback, published 26 November 2020

278 pages, 10 b/w illus. 3 colour illus. 45 exercises
23.5 x 15.7 x 1.8 cm, 0.54 kg

'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

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.'

A non-technical introduction
1. A guided tour through the land of operator semigroups
2. Generators versus intensity matrices
3. Boundary theory: core results
4. Boundary theory continued
5. The dual perspective
Solutions and hints to selected exercises
Commonly used notations
References
Index.

Subject Areas: Stochastics [PBWL], Functional analysis & transforms [PBKF]