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Levy Processes, Integral Equations, Statistical Physics: Connections and Interactions

  • Book
  • © 2012

Overview

Authors:
  1. Lev A. Sakhnovich

    1. Milford, USA

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  • Investigation of the interconnection between probability problems and analysis problems
  • Consideration of the statistical problems using the game theory ideas
  • Construction of special examples instead of well-known existence theorems
  • Generalization and investigation of the notion of integrable operators
  • Includes supplementary material: sn.pub/extras

Part of the book series: Operator Theory: Advances and Applications (OT, volume 225)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-ix
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  2. Levy processes

    • Lev A. Sakhnovich
    Pages 11-51

  3. The principle of imperceptibility of the boundary in the theory of stable processes

    • Lev A. Sakhnovich
    Pages 53-62

  4. Approximation of positive functions by linear positive polynomial operators

    • Lev A. Sakhnovich
    Pages 63-75

  5. Optimal prediction and matched filtering for generalized stationary processes

    • Lev A. Sakhnovich
    Pages 77-84

  6. Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

    • Lev A. Sakhnovich
    Pages 85-99

  7. Comparison of thermodynamic characteristics of quantum and classical approaches

    • Lev A. Sakhnovich
    Pages 101-112

  8. Dual canonical systems and dual matrix string equations

    • Lev A. Sakhnovich
    Pages 113-149

  9. Integrable operators and canonical differential systems

    • Lev A. Sakhnovich
    Pages 151-167

  10. The game between energy and entropy

    • Lev A. Sakhnovich
    Pages 169-179

  11. Inhomogeneous Boltzmann equations: distance, asymptotics and comparison of the classical and quantum cases

    • Lev A. Sakhnovich
    Pages 181-199

  12. Operator Bezoutiant and roots of entire functions, concrete examples

    • Lev A. Sakhnovich
    Pages 201-221

  13. Back Matter

    Pages 223-245
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Keywords

About this book

In a number of famous works, M. Kac showed that various methods of probability theory can be fruitfully applied to important problems of analysis. The interconnection between probability and analysis also plays a central role in the present book. However, our approach is mainly based on the application of analysis methods (the method of operator identities, integral equations theory, dual systems, integrable equations) to probability theory (Levy processes, M. Kac's problems, the principle of imperceptibility of the boundary, signal theory). The essential part of the book is dedicated to problems of statistical physics (classical and quantum cases). We consider the corresponding statistical problems (Gibbs-type formulas, non-extensive statistical mechanics, Boltzmann equation) from the game point of view (the game between energy and entropy). One chapter is dedicated to the construction of special examples instead of existence theorems (D. Larson's theorem, Ringrose's hypothesis, the Kadison-Singer and Gohberg-Krein questions). We also investigate the Bezoutiant operator. In this context, we do not make the assumption that the Bezoutiant operator is normally solvable, allowing us to investigate the special classes of the entire functions.

Authors and Affiliations

  • Milford, USA

    Lev A. Sakhnovich

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