We consider Markov processes, with countable set of states, interpreted as systems of particles of several types that interact as complexes for which the result of interaction with a complex of particles does not depend on the presence of other particles in the system. The apparatus of multivariate generating functions is used to find exact closed solutions of the first and second Kolmogorov system of differential equations for the transition probabilities. In our examples analytic methods are used to consider actual transmutation processes of particles in diverse domains of science.
The transition probabilities of Markov processes with countable state space satisfy the first and second systems of differential equations, which are linear. The Markov branching process introduced in [Kolmogorov A.N., Dmitriev N.A. Random branching processes. Doklady Akademii Nauk SSSR, 1947, v. 56, no. 1, p. 7-10. (In Russian.)] was described as a process of evolution of particles; under the assumption that the individual evolving particles are independent of one another, a non-linear first-order differential equation was obtained in [DAN, 1947] for the generating functions of transition probabilities of such a process. The authors of the paper stressed: " ... the remark shows that our ’branching stochastic processes’ are in fact only a special case of Markov processes with countable state space. However, we shall obtain an analytic apparatus for this special case which is much more efficient than the apparatus that can be developed for the general case of Markov processes with countable state space" (the accentuation in the text is due to the authors of [DAN, 1947]). Thus, the paper [DAN, 1947] poses the following questions. Are there other special cases of Markov processes with countable state space whose transition probabilities satisfy a non-linear equation? If there are examples of Markov processes of this kind, then how to single out possible special classes of Markov processes whose transition probabilities satisfy non-linear equations of diverse types in the set of all Markov processes?
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Preprints
- David Sirl in the Bauman University
- Andrey Lange in the University of Queensland, University of Melbourne, Monash University (Australia) and Chalmers University of Technology, University of Gothenburg (Sweden)