Pogoruy A. Systems of interacting particles in Markov media

This thesis is devoted to stochastic processes with Markov and semi-Markov switching times that model both the motion of a particle in a multidimensional space and interacting particles systems on the line with Markov switching. A telegraph-type equation is introduced for obtaining the pdf of the position of a particle moving along a line with velocities driven by a switching process having a generalized Erlang distribution. A method for solving these partial differential equations by means of monogenic functions on the associated commutative algebras is developed. It is obtained an explicit form for the distribution of the particle's limiting position of fading evolutions on the line in the cases of uniform and Erlang distributions of switching processes. It is proved that the characteristic function of a particle moving in a multidimensional space with random velocities driven by the general renewal process satisfies the integral equation of convolution. Geometrical characteristics of systems of interacting particles with Markov switching and their asymptotic behavior as time tends to infinity are studied.