Random Evolutionary Systems

3Large Deviations in the Scheme of Asymptotically Small Diffusion

This chapter studies the asymptotic analysis of the large deviations problem for random evolutionary systems in the scheme of asymptotically small diffusion. The theory of large deviations deals with the asymptotic estimation of probabilities of rare events. The method, used in the majority of classical works, is based on the change of measure and the application of the variational formula to the cumulant of the process under study. The method of asymptotic analysis of the exponential generator of the Markov process is used. The limit exponential generators are calculated for random evolutionary systems with ergodic Markov switching. The method proposed here may have applications for finite-dimensional models, such as random evolutionary systems in R^d, queuing theory, etc.

3.1. Statement of the problem

The theory of large deviations was presented in the work of Cramér (1938), which deals with the asymptotic estimation of probabilities of rare events. The main problem in the large deviations theory is the construction of the rate functional to estimate the probabilities of rare events. Different aspects and applications of this problem were studied by many mathematicians. We discuss the Markov processes with independent increments, so it is natural to refer to the fundamental works of Donsker and Varadhan (1975a, 1975b, 1976), Ventsel (1976a, 1976b, 1979) and Freidlin and Ventzell (2012).

Another approach was ...