ABSTRACT
In this chapter, the author gives a detailed analysis of the symmetric Markov random flight, in the four-dimensional Euclidean space. Surprisingly, despite fairly high dimension, one managed to obtain an explicit distribution in terms of elementary functions that in itself is a very rare result. The author obtains closed-form expressions for the conditional densities that, surprisingly, have very simple forms and derives exact formula for the distribution of process, which is expressed in terms of elementary functions basing it on the obtained conditional densities. The general model of symmetric Markov random flight in the four-dimensional case is represented by the stochastic motion of a particle that, at the initial time moment, starts from the origin of the Euclidean space and moves with a constant speed. Conditional distributions enables us to immediately derive the absolutely continuous part of the distribution.
