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.