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, 2004), 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 (1986), Swishchuk (1989), Hersh and Papanicolaou (1972), Iksanov and Rösler (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. (1993a) 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 (2005) 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 (2011) 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
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 (2011) 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 (2011) 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 been studied by Qian et al. (1998), Cahoy (2007), Beghin and Orsingher (2010b), Orsingher and Beghin (2009), D’Ovidio et al. (2014), and others.
The set of particles with interaction, where each particle moves on a line according to a telegraph process, up to collision with another particle, was studied by Pogorui (2012b). During the collision, the particles exchange momentums. In this book, the author calculates the distribution of time of the first collision for two telegraph particles that started simultaneously from different points on a line and investigates the limit of this distribution under Kac’s condition. The author also investigates the system of particles with Markov switching, which is bounded with reflecting boundaries. The distribution for the position of particles of the system in a fixed time was also obtained. The limiting properties of these distributions and an estimate of the number of collisions in the system with reflecting boundaries, as well as without them, are also studied. Such a system of particles can be interpreted as a model of one-dimensional gas and it is a kind of one-dimensional generalization of the deterministic models of gas, of the billiard type, that were studied by Kornfeld et al. (1982), for example. The velocity of particles in these models is considered to be finite. This is a major difference from systems where the position of a particle is described by a diffusion process, such as in Arratia flow. We should note that models with finite speeds of particles moving under the influence of forces of mutual attraction were studied by Sinai (1992), Lifshits and Shi (2005), Giraud (2001, 2005), Bertoin (2002) and Vysotsky (2008a).
I.2. Description of the book
The book is divided into two volumes, each containing two parts. Part 1 of Volume 1 consists of basic concepts and methods developed for random evolutions. These methods are the elementary tools for the rest of the book, and they include many results in potential operators and the description of some
