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.
