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      Introduction

      I.1. Overview

      The theory of dynamical systems is one of the fields of modern mathematics that is under intensive study. In the theory of stochastic processes, researchers are actively studying dynamical systems, operating under the influence of random factors. A good representative of such systems is the theory of random evolutions. The first results within this field were obtained by Goldstein (1951) and Kac (1974), who studied the movement of a particle on a line with a speed that changes its sign under the Poisson process. Subsequently, this process was called the telegraph process or the Goldstein–Kac process. Further developments of this theory have been presented in the works of Griego and Hersh (1969, 1971), Hersh and Pinsky (1972), and Hersh (1974, 2003), which gave a definition of stochastic evolutions in a general setting.

      Important advances in the theory of stochastic evolutions have been made in the formulation of limit theorems and their refinement; these consist of obtaining asymptotic expansions. These problems are studied in the works of Korolyuk and Turbin (1993), Skorokhod (1989), Turbin (1972, 1981), Korolyuk and Swishchuk (1986), Korolyuk and Limnios (2009, 2005), Shurenkov (1989, 1986), Dorogovtsev (2007b), Anisimov (1977), Girko (1982), Hersh and Pinsky (1972), Papanicolaou (1971a,b), Kertz (1978), Watkins (1984, 1985), Balakrishnan et al. (1988), Yeleyko and Zhernovyi (2002), Pogorui (1989, 1994, 2009a) and Pogorui and Rodríguez-Dagnino (2006, 2010a).

Among the many methods in finding limiting theorems, we should mention a class that could be called an asymptotic average scheme. By using these methods, a semi-Markov evolution can be reduced to a random evolution in a lumping state space with Markov switching, and is thus studied as a Markov scheme. Most of the results in this field are presented in the following books and papers: Korolyuk and Swishchuk (1995b), Korolyuk and Korolyuk (1999), Korolyuk and Turbin (1993), Korolyuk and Limnios (2005), Turbin (1972), Pogorui (2004a, 2012a), and Rodríguez-Said et al. (2007). Furthermore, this theory was also developed in the study of asymptotic expansions for functionals of random evolution in the phase averaging and diffusion approximation. This topic was the main subject of the following works: Korolyuk and Limnios (2009, 2004), Turbin (1981), Samoilenko (2005), Albeverio et al. (2009), Pogorui (2010a), Pogorui and Rodríguez-Dagnino (2010a), and Nischenko (2001).

      The asymptotic average scheme has been applied to a semi-Markov evolution for the computation of the effectiveness of a multiphase system with a couple of storage units by Pogorui (2003, 2004a), Pogorui and Turbin (2002), and Rodríguez-Said et al. (2007).

      Research on random evolutions has also been carried out by applying martingale methods. It seems that the founders of this approach were Stroock and Varadhan (1969, 1979), but further developments can be found in the works of Skorokhod (1989), Pinsky (1991), Korolyuk and Korolyuk (1999), Sviridenko (1989), Swishchuk (1989), Hersh and Papanicolaou (1972), Iksanov and Résler (2006), and Griego and Korzeniowski (1989).

      In addition to the successful development of abstract stochastic evolutions in the decade 1980–1990, several scholars studied various generalizations of the Goldstein–Kac telegraph process to multidimensional spaces. In connection to this, the results of Gorostiza (1973), Gorostiza and Griego (1979), Orsingher (1985), Orsingher and Somella (2004), Turbin (1998), Orsingher and Ratanov (2002), Samoilenko (2001) and Lachal (2006) should be noted. In most of these works, the authors considered a finite number of directions of a particle movement and obtained differential equations for the probability density functions. In the works of Pogorui (2007) and Pogorui et al. (2014), the authors proposed a method for solving such equations by using monogenic functions, after associating a particular commutative algebra with them.

      Masoliver et al. (1993) studied the telegraph process with reflecting or partially reflecting boundaries and its distribution in a fixed interval of time. In papers by Pogorui (2005, 2006), and Pogorui and Rodríguez-Dagnino (2006, 2010b), the authors also study the stationary distribution of some Markov and semi-Markov evolution with delaying boundaries.

      De Gregorio et al. (2005), and Stadje and Zacks (2004) considered the generalizationof the telegraph process on a line, where there is a discrete set of particle velocities, then at Poisson epochs a velocity is chosen from this set.

      Pogorui (2010b), Pogorui and Rodríguez-Dagnino (2009b) and Samoilenko (2002) investigated fading evolutions, where the velocity of a particle tends to zero as the number of switches grows at infinite.

      Pogorui and Rodríguez-Dagnino (2005a) also studied a generalization of the telegraph process to the case of Erlang interarrivals between successive switches of particle velocities. For such processes, a differential equation for the probability density function (pdf) of the particle position on the line was obtained. In addition, in Pogorui (2011a) a method for solving such equations, by using monogenic functions associated with this equation on a commutative algebra, was developed.

      Orsingher and De Gregorio (2007), Stadje (2007) and others studied the motion of a particle in multidimensional spaces with constant absolute velocity and directions uniformly distributed on a unit sphere that change at Poisson points. The authors obtained explicit formulas for the position of the particle distributions for two- and four-dimensional space and investigated the “explosive effect” for the pdf of the position of a particle that approaches the singularity sphere for the plane and the three-dimensional space.

      In recent years, much attention has been paid to the Pearson random walk with Gamma distributed steps and associated walks of Pearson–Dirichlet type, whose steps have the Dirichlet distribution. Franceschetti (2007) developed explicit formulas for the conditional pdf of the position of a particle at any number of steps for a walk in ℝⁿ (n = 1, 2) with uniformly distributed directions and steps with the Dirichlet distribution with parameter q = 1. Beghin and Orsingher (2010a) obtained an expression for the conditional distribution of the position of a particle of the Pearson–Dirichlet walk with parameter q = 2 in the plane. Le Caér (2010, 2011) generalized these results to the case of the multidimensional Pearson–Dirichlet random walk with arbitrary parameter q, where the author introduces the concept of a “Hyperspherical Uniform” (HU) random walk. The HU walk is a motion, the endpoint distribution of which is identical to the distribution of the projection in the walk space of a point, with a position vector, randomly chosen on the surface of the unit hypersphere of some hyperspace in higher dimensions. By using properties of the HU random walk, Le Caër found walks for which the conditional probability density can be expressed in a closed form. We should also mention the recent papers by De Gregorio (2014) and Letac and Piccioni (2014), where they obtained a generalization and simplification of proofs of the results stated by Le Caër.

In all of the above-mentioned papers, the authors study the conditional distributions of the particle position at renewal epochs of the switching direction process. For a non-Markov switching process, it is possible to find the corresponding results. The mathematical technique is to replace the distribution of the non-Markov process by the Markov chain walk embedded in this process. By using this approach, Pogorui and Rodríguez-Dagnino (2011a) obtained a recursive expression for the conditional characteristic functions of a random walk with Erlang switching, considering a non-Markov switching process. Namely, they studied the changes in the conditional characteristic functions of the particle position, not only at instants of the direction changing, but at all Poisson times. Pogorui (2011a) studied the particular case of the Erlang distributed stay of the switching process in the states. Further results in this direction were obtained by Pogorui and Rodríguez-Dagnino (2012) where the authors also studied multidimensional random motion at random velocities. For some distributions of random velocity, they observed an “explosive effect” for the pdf of the position of a particle when it is approaching the singularity sphere in four-dimensional space.

      Other directions of random walk theory, which have been studied intensively during recent years, are the fractal Brownian motion and the fractal generalization of the telegraph process. These processes have

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