Random Evolutions and Their Applications

1. Front Matter

   Pages i-xvi

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2. Introduction

   • Anatoly Swishchuk

   Pages 1-5

3. Multiplicative Operator Functionals

   • Anatoly Swishchuk

   Pages 7-18

4. Random Evolutions

   • Anatoly Swishchuk

   Pages 19-60

5. Limit Theorems for Random Evolutions

   • Anatoly Swishchuk

   Pages 61-102

6. Applications of Evolutionary Stochastic Systems

   • Anatoly Swishchuk

   Pages 103-147

7. New Trends in Random Evolutions

   • Anatoly Swishchuk

   Pages 149-178

8. Back Matter

   Pages 179-200

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The main purpose of this handbook is to summarize and to put in order the ideas, methods, results and literature on the theory of random evolutions and their applications to the evolutionary stochastic systems in random media, and also to present some new trends in the theory of random evolutions and their applications. In physical language, a random evolution ( RE ) is a model for a dynamical sys­ tem whose state of evolution is subject to random variations. Such systems arise in all branches of science. For example, random Hamiltonian and Schrodinger equations with random potential in quantum mechanics, Maxwell's equation with a random refractive index in electrodynamics, transport equations associated with the trajec­ tory of a particle whose speed and direction change at random, etc. There are the examples of a single abstract situation in which an evolving system changes its "mode of evolution" or "law of motion" because of random changes of the "environment" or in a "medium". So, in mathematical language, a RE is a solution of stochastic operator integral equations in a Banach space. The operator coefficients of such equations depend on random parameters. Of course, in such generality , our equation includes any homogeneous linear evolving system. Particular examples of such equations were studied in physical applications many years ago. A general mathematical theory of such equations has been developed since 1969, the Theory of Random Evolutions.

• Operator theory
• Optimal control
• Probability theory
• functional analysis
• statistics

• Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

  Anatoly Swishchuk

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