Random Evolutionary Systems

6Difference Diffusion Models with Equilibrium

6.1. Statistical experiments with linear persistent regression

In physical processes with equilibrium, the dynamics of concentration, or frequencies, of a predefined characteristic, can be described by a mathematical model of binary statistical experiments, based on the statistical data of the elementary hypothesis validation, about the presence or absence of a predefined attribute A in the set of elements that make up a complex system.

It is assumed that:

• – all of the elements that make up the system can gain or lose the attribute A over time, i.e. the frequency of attribute A is a dynamic variable;
• – the basic objects of our study are the statistical experiments, characterized by the relative frequencies of the presence or absence of the attribute A, in a sample of fixed volume at each time instant;
• – it is assumed that the (average) results of the next experiment (at time instant k + 1) depend on average result of the present experiment at time instant k, and do not depend on all previous time instants k − 1, k − 2, etc. This relationship is called the feature of persistent regression and is used as a fundamental condition for the subsequent analysis of the model.

The method of constructing and exploring the proposed mathematical model is based on the analysis of the following basic properties of statistical experiments:

• – persistent regression;
• – equilibrium value and fluctuations, as well as their asymptotic behavior;