Mathematical Foundations of the State Lumping of Large Systems

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]).

Inhalt

1. Classes of Linear Operators.- 1.1. Basic notions.- 1.2. Closed and closable operators.- 1.3. Normally solvable operators.- 1.4. Invertibly reducible operators.- 1.5. Pseudo-resolvents.- 2. Semigroups of Operators and Markov Processes.- 2.1. Basic notions.- 2.2. Infinitesimal operators of ergodic Markov processes.- 2.3. Holomorphic semigroups with invertibly reducible infinitesimal operators.- 2.4. Semigroups of operators uniformly and strongly ergodic at the infinity.- 2.5. Generating operators of ergodic semi-Markov processes.- 2.6. Abstract potential operators.- 2.7. Examples of invertibly reducible operators.- 3. Perturbations of Invertibly Reducible Operators.- 3.1. Eigen-projectors and eigen-operators.- 3.2. Inversion of an invertibly reducible operator perturbed on the spectrum.- 3.3. Resolvents of singularly perturbed semigroups.- 3.4. Limit theorems and asymptotic expansions for resolvents of singularly perturbed semigroups.- 3.5. Limit theorems and asymptotic expansions for resolvents of singularly perturbed semigroups. The case of s > 2.- 4. Singular Perturbations of Holomorphic Semigroups.- 4.1. Principal problems. The method of Vishyk-Lyusternik-Vasilyeva.- 4.2. Structure of singularly perturbed semigroups.- 4.3. Regular lumped approximations to solutions of singularly perturbed equations.- 5. Asymptotic Expansions and Limit Theorems.- 5.1. Strong limits of singularly perturbed semigroups. Resolvent approach.- 5.2. Asymptotic analysis of singularly perturbed semigroups. The case of s=1.- 5.3. Asymptotic analysis of singularly perturbed semigroups.- 6. Asymptotic Phase Lumping of Markov and Semi-Markov Processes.- 6.1. Limit theorems.- 6.2. Asymptotic phase lumping. The case of s=1.- 6.3. Some examples.- 6.4. Asymptotic phase lumping. The case of s? 2.- 6.5. Classification of processes admitting asymptotic phase lumping.- 6.6. Limit theorems and asymptotic theorems for additive functionals.- 7. Applications of the Theory of Singularly Perturbed Semigroups.- 7.1. Tikhonov systems of differential equations.- 7.2. Nonrelativistic limit of the Dirac operator.- 7.3. Hydrodynamic limit for the linearized Boltzmann equation.- References.