Random Evolutionary Systems

7Random Evolutionary Systems in Discrete–Continuous Time

7.1. Discrete Markov evolutions in an asymptotic diffusion environment

The asymptotic diffusion environment, defined by a solution of a difference evolutionary equation in an ergodic Markov environment, determines the limit diffusion environment for a discrete Markov evolution.

7.1.1. Asymptotic diffusion perturbation

The Markov random environment is given by a two-component Markov chain (x_n, τ_n), n ≥ 0, homogeneous by the second component (Korolyuk and Limnios 2005):

[7.1]

The semi-Markov kernel

[7.2]

specifies the transition probabilities of a two-component Markov chain (x_n, τ_n), n ≥ 0:

[7.3]

[7.4]

Here, the random variables

[7.5]

determine the sojourn time in the states x ∈ E of the embedded Markov chain x_n, n ≥ 0. In the particular case of exponentiality of distributions [7.4]

The Markov random environment is really Markov, ...