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

Abstract

A family of one-dimensional continuous-time Markov processes is considered, for which the author has earlier determined the transition probabilities by directly solving the Kolmogorov–Chapman equation; these probabilities have the form of single integrals. Analogues of the first and second Kolmogorov equations for the family of processes under consideration are obtained by using a procedure for obtaining integro-differential equations describing Markov processes with discontinuous trajectories. These equations turn out to be equations in fractional derivatives. The results are based on an asymptotic analysis of the transition probability as the start and end times of the transition approach each other. This analysis implies that the trajectories of a given Markov process are divided into two classes, depending on the interval in which they start. Some of the trajectories decay during a short time interval with a certain probability, and others are generated with a certain probability.