Random Evolutionary Systems

1Basic Tools for Asymptotic Analysis

1.1. Basic concepts of operator asymptotic analysis

This first chapter introduces the known auxiliary concepts, which are actively used in the following sections. In particular, the concepts of the Poisson approximation and Lévy approximation, impulsive recurrent process, random evolutionary system with locally independent increments, compensating operator, Nisio semigroup, nonlinear exponential operator, martingale control problem, rate functional, compactness, etc. are introduced. The following results are new:

• 1. The previously known results were used for a detailed substantiation of the relationship between the Nisio semigroup, the martingale control problem, the Brick formula, the nonlinear exponential operator and the corresponding rate functional. Some comparative examples are given for the classical formulation of the problem of large deviations in the scheme of small diffusion.
• 2. The problems related to the study of compactness and exponential compactness of processes, as well as the application of the comparison condition to the limit nonlinear exponential generator are described. For the processes that will be studied further, the corresponding compactness conditions are tested using the methods from the stability theory, in particular the Lyapunov function. For the corresponding nonlinear exponential generator, the comparison condition is checked.
• 3. For the first time, the problem of studying the problem of large deviations ...