Kolmogorov equations in fractional derivatives for the transition probabilities of some Markov processes with continuous time

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.