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Generators of Markov Chains

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

Part of Cambridge Studies in Advanced Mathematics

  • Publication planned for: January 2021
  • availability: Not yet published - available from January 2021
  • format: Hardback
  • isbn: 9781108495790

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  • 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 focuses on time-continuous chains but is not so far from being elementary itself. It reminds us once again that the first 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.'

    • Takes a much simpler approach than the existing literature
    • Encourages the reader to discover the facts for themselves by examining examples before learning the theorem
    • Contains unusual, fascinating examples of Markov chains, gathered from the works of Blackwell, Feller, Kolmogorov, Kendall, Lévy and Reuter
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    Reviews & endorsements

    '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

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    Product details

    • Publication planned for: January 2021
    • format: Hardback
    • isbn: 9781108495790
    • dimensions: 228 x 152 mm
    • contains: 10 b/w illus. 3 colour illus. 45 exercises
    • availability: Not yet published - available from January 2021
  • Table of Contents

    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.

  • Author

    Adam Bobrowski, Lublin University of Technology
    Adam Bobrowski is a professor and Chairman of the Department of Mathematics at Lublin University of Technology, Poland. He is a pure mathematician who uses the language of operator semigroups to describe stochastic processes. He has authored and co-authored nearly 70 papers on the subject, and five books, including Functional Analysis for Probability and Stochastic Processes (2005) and Convergence of One-Parameter Operator Semigroups (2016).

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