Kolmogorov equations in fractional derivatives for the transition probabilities of some Markov processes with continuous time
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St Petersburg State University
Abstract
We consider a family of one-dimensional Markov processes with continuous time, for which earlier the
author received the transition probability by means of the Chapman Kolmogorov equation. These
probabilities have the form of simple integrals. Using the procedure for obtained integral-differential
equations for Markov processes with discontinuous trajectories, the author gets both first and second
Kolmogorov equations for this family of processes. These equations are called equations with fractional
derivatives. Results are based on the asymptotic analysis of transition probability when approaching the
start of transition and the time of the end. From this analysis,in particular, it is followed the trajectory of
Markov process be divided into two classes according to the range where they started. Some trajectories
disappear with a certain probability, while others are born with a certain probability. Refs 8.
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Miroshin R.N. Kolmogorov equations in fractional derivatives for the transition probabilities of some Markov processes with continuous time. Vestnik SPbSU. Mathematics. Mechanics. Astronomy, 2017, vol. 4 (62), issue 1, pp. 38–48.