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

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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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.

• Levy processes
• energy
• entropy
• stable processes
• triangular factorization
• combinatorics

• Milford, USA

  Lev A. Sakhnovich

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