Kinetic Theory and Markov Chains with Stochastically Varying Transition Matrices

As is well-known, the Kinetic Theory for Active Particles is a scheme of mathematical models based on a generalization of the Boltzmann equation. It must be nowadays acknowledged as one of the most versatile and effective tools to describe in mathematical terms the behavior of any system consisting of a large number of mutually interacting objects, no matter whether they also interact with the external world. In both cases, the description is stochastic, i.e., it aims to provide at each instant the probability distribution (or density) function on the set of possible states of the particles of the system. In other words, it describes the evolution of the system as a stochastic process. In a previous paper, we pointed out that such a process can be described in turn in terms of a special kind of vector time-continuous Markov Chain. These stochastic processes share important properties with many natural processes. The present paper aims to develop the discussion presented in that paper, in particular by considering and analyzing the case in which the transition matrices of the chain are neither constant (stationary Markov Chains) nor assigned functions of time (nonstationary Markov Chains). It is shown that this case expresses interactions of the system with the external world, with particular reference to random external events.