Stabilization of Distribution of Markov's Process in Wide Sense
by
Sergey N. Gerasin
PhD, Assosiated Professor,61166,Lenin av.14,Mathematics Department, Kharkov Technikal University of Radioelectronics,Kharkov,Ukraine

The problem about focusing and stabilization of Markov's process states probabilities in wide sence are discussed in this report. Suppose that a set of elementary results \Omega is Rⁿ or it is it's subset.The set of events consist of all borel set B_i in \Omega everywhere dense in \Omega.On all set B_i probably measure is defined. Focusing of probability we'll call that evolution of process in which probability are reach the given values during the final period of time. If in final period of time the probabilities of states are localized in the small neighberhood of components of the given distribution then in this case we can speak about stabilization of process. Let's consider stochastic matrix of process P_st=|| P_st(B_i, B_j)|| which elements are probabilities of transition between B_i and B_j in suggestion that the process situated in B_i at the moment of time s and runs to the state B_j at the moment t.Focusing and stabilization at the moment t₀ may be realized if some special chosen elements of matrix P_st will be strongly perturbated (or all matrix elements or their part ). This perturbation must provide implementation of the conditions:

1. Suppose that such an sequence of indexes j_k and such an sequence

exist that

2. An sequence of eigenvectors [( --> ) || P](t) of matrix P_st ([( --> ) || P](t)P_st=[( --> ) || P](t)) has a limit

Date received: March 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadz-80.