Mathematical Foundations of the State Lumping of Large Systems

ISBN-13: 9789401049191 / Angielski / Miękka / 2012 / 278 str.

Vladimir S. Korolyuk; A. F. Turbin

Opis

During the investigation of large systems described by evolution equations, we encounter many problems. Of special interest is the problem of "high dimensionality" or, more precisely, the problem of the complexity of the phase space. The notion of the "comple- xity of the. phase space" includes not only the high dimensionality of, say, a system of linear equations which appear in the mathematical model of the system (in the case when the phase space of the model is finite but very large), as this is usually understood, but also the structure of the phase space itself, which can be a finite, countable, continual, or, in general, arbitrary set equipped with the structure of a measurable space. Certainly, 6 6 this does not mean that, for example, the space (R 6, ( ), where 6 is a a-algebra of Borel sets in R 6, considered as a phase space of, say, a six-dimensional Wiener process (see Gikhman and Skorokhod [1]), has a "complex structure". But this will be true if the 6 same space (R 6, ( ) is regarded as a phase space of an evolution system describing, for example, the motion of a particle with small mass in a viscous liquid (see Chandrasek- har [1]).

Spis treści

Introduction. 1. Classes of Linear Operators. 2. Semigroups of Operators and Markov Processes. 3. Perturbations of Invertibly Reducible Operators. 4. Singular Perturbations of Holomorphic Semigroups. 5. Asymptotic Expansions and Limit Theorems. 6. Asymptotic Phase Lumping of Markov and Semi-Markov Processes. 7. Applications of the Theory of Singularly Perturbed Semigroups. References. Subject Index.