Random Evolutionary Systems

4Large Deviations of Systems in Poisson and Lévy Approximation Schemes

4.1. Random evolutionary systems with independent increments

4.1.1. Poisson approximation scheme

Consider the large deviations problem for random evolutionary systems in the Poisson approximation scheme (see section 1.1.10) with Markov switching (see section 1.1.3) from the point of view of the program proposed in section 1.2. Note that the second and third stages of this program are implemented in section 1.3, therefore, we will focus on the asymptotic analysis of the nonlinear process generator and the calculation of the rate functional by applying convex analysis methods.

V.S. Korolyuk (2010a, 2010b) proposed applying a solution of the singular perturbation problem in the study of large deviations for random evolutionary systems with independent increments in the scheme of asymptotically small diffusion.

In classical works, asymptotic analysis of the large deviations problem is usually performed using a large series parameter n → ∞, sometimes several different parameters (see, for example, Mogulskii (1993)).

The normalization of random evolutionary systems by a small series parameter in order to solve the large deviations problem in the Poisson approximation scheme is performed using two small parameters (δ normalizes the intensity of jumps of the process, ε normalizes the time and the intensity of jumps):

where ε, δ → 0, hence ε⁻¹δ → 1.

Let us rewrite the conditions of the Poisson approximation according ...