On the asymmetric telegraph processes Oscar L ´ opez and Nikita Ratanov Universidad del Rosario, Cl. 12c, No. 4-69, Bogot´ a, Colombia Email address: oscar.lopez@ur.edu.co; nratanov@urosario.edu.co Abstract We study the one-dimensional random motion X = X (t ), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formu- lae for the density functions of X and for the moments of any order as well as the distributions of the first passage times are obtained. The limit behaviour of the mo- ments is analysed under non-standard Kac’s scaling. Keywords: asymmetric telegraph process; first passage times; moments; Kac’s asymp- totics; modified Bessel functions; Kummer function Published in JOURNAL OF APPLIED PROBABILITY 51(2) JUNE 2014 1 Introduction The main subject of study in this paper is the mathematical description of non-interacting particles moving in one dimension with alternating finite velocities, the so-called telegraph (or telegrapher’s) stochastic process. Beginning with 1956 lecture notes by M.Kac [13] the telegraph processes and their numerous generalisations have studied in great detail, see, e. g., [2], [4], [5], [7], [18], [19], [20], [21], [25] with applications in physics [24], biology [10], [11], ecology [17] and, more recently, in financial market modelling [22] (see also the bibliography in these papers). In the symmetric case, i. e. if the particles move with symmetric velocities ±c, switch- ing the directions with the intensity λ , λ > 0, the density function p = p(x, t ) of particles’ positions satisfies the hyperbolic differential equation of the second order, the telegraph (or damped wave) equation, see [13]: ∂ 2 p(x, t ) ∂ t 2 + 2λ ∂ p(x, t ) ∂ t = c 2 ∂ 2 p(x, t ) ∂ x 2 , t > 0. (1.1) Under scaling condition, c, λ → ∞, such that c 2 /λ → σ 2 , proposed by M.Kac, equation (1.1) converges to the heat equation and the dynamical particles’ distribution weakly con- verges to the diffusion with the diffusion coefficient σ 2 . If the velocities are different in different directions, the particles’ movement holds only an additional drift. In this paper we assume that the movement is asymmetric not only with 1