Large Deviations for Random Evolutions in the Scheme of Asymptotically Small Diffusion
The theory of large deviations deals with the asymptotic estimations for probabilities of rare events. The method, used in majority of classical works, is based on the change of measure and application of the variational formula to the cumulant of the process under study. Here the large deviations problem is considered for random evolutions in the scheme of asymptotically small diffusion. The method of asymptotic analysis for the exponential generator of the Markov process is used. The limit exponential generators are calculated for random evolution with the ergodic Markov switching (Sect. 3) and with the split-and-double merging switching Markov process (Sect. 4). The method proposed here may have applications for the finite dimensional models arising in the theory of random evolutions in R d , queuing theory, etc.
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