Random Evolutionary Systems

5Large Deviations of Systems in the Scheme of Splitting and Double Merging

As we have already mentioned, the classical formulations of the problem in the study of large deviations relate to the small diffusion scheme. A problem that is similar to the Poisson approximation scheme in the case of processes with independent increments was investigated in Mogulskii (1993).

In this section, we describe how the approach proposed in Chapters 3 and 4 can be applied in cases that could not be studied using classical methods. That is, we describe the situation when the switching Markov process has a complicated structure. We introduce the switching Markov process x^ε(t), t ≥ 0 on the standard phase (state) space (E, ℰ) in the series scheme with a small series parameter ε → 0, ε > 0 on the split phase space

The Markov kernel is

The problem of studying large deviations in the phase merging scheme (see section 1.1.5 and conditions ME1–ME4), both in the small diffusion scheme and the Poisson approximation scheme, is new and still unexplored. It should be noted that an attempt to set such a task was made in Mogulskii (1993), but its formulation and method of solution do not meet the stated goal.

5.1. Small diffusion scheme

5.1.1. Large deviations under the local balance (LB) condition

Random ...