First-passage time, maximum displacement, and Kac's solution of the telegrapher equation

Reflection principle for finite-velocity random motions

We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

First-passage probabilities and mean number of sites visited by a persistent random walker in one- and two-dimensional lattices

We study the first passage probability and mean number of sites visited by a discrete persistent random walker on a lattice in one and two dimensions. This is performed by using the multistate formulation of the process. We obtain explicit expressions for the generating functions of these quantities. To evaluate these expressions, we study the system in the strongly persistent limit. In the one-dimensional case, we recover the behavior of the continuous one-dimensional persistent random walk (telegrapher process). In two dimensions we obtain an explicit expression for the probability distribution in the strongly persistent limit, however, the Laplace transforms required to evaluate the first-passage processes could only be evaluated in the asymptotic limit corresponding to long times in which regime we recover the behavior of normal two-dimensional diffusion.

Linear combinations of the telegraph random processes driven by partial differential equations

Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients.

We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.

Telegraph processes with random velocities

We study a one-dimensional telegraph process (Mt)t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of Mt and its first two moments are derived. We characterize the level hitting times of Mt in terms of integro-differential equations which can be solved in special cases.

Generalized integrated telegraph processes and the distribution of related stopping times

Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

A Damped Telegraph Random Process with Logistic Stationary Distribution

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

Generalized integrated telegraph processes and the distribution of related stopping times

Let {X(t),V(t),t≥ 0} be a telegraph process, withV(0+) = 1. The distribution ofX(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levelsaand −aare studied for M/G and for G/M processes.

Telegraph processes with random velocities

We study a one-dimensional telegraph process (Mt)t≥0describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution ofMtand its first two moments are derived. We characterize the level hitting times ofMtin terms of integro-differential equations which can be solved in special cases.

A Damped Telegraph Random Process with Logistic Stationary Distribution

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

On random motions with velocities alternating at Erlang-distributed random times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

On random motions with velocities alternating at Erlang-distributed random times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line ℝ. The processes Xk(t), k = 1, 2, describe stochastic motions at finite constant velocities c1 > 0 and c2 > 0 that start at the initial time instant t = 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1 > 0 and λ2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) - X2(t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processesX1(t) andX2(t) on the real line ℝ. The processesXk(t),k= 1, 2, describe stochastic motions at finite constant velocitiesc1> 0 andc2> 0 that start at the initial time instantt= 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1> 0 and λ2> 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) -X2(t)|,t> 0, between these processes at an arbitrary time instantt> 0. Some numerical results are also presented.

On the Asymmetric Telegraph Processes

We study the one-dimensional random motionX=X(t),t≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions ofXand for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

The explicit probability distribution of the sum of two telegraph processes

We consider two independent Goldstein–Kac telegraph processes X1(t) and X2(t) on the real line ℝ, both developing with constant speed c > 0, that, at the initial time instant t = 0, simultaneously start from the origin 0 ∈ ℝ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ > 0. Closed-form expressions for the transition density φ(x, t) and the probability distribution function Φ(x, t) = Pr {S(t) < x}, x ∈ ℝ, t > 0, of the sum S(t) = X1(t) + X2(t) of these processes at arbitrary time instant t > 0, are obtained. It is also proved that the shifted time derivative g(x, t) = (∂/∂t + 2λ)φ(x, t) satisfies the Goldstein–Kac telegraph equation with doubled parameters 2c and 2λ. From this fact it follows that φ(x, t) solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.

On the Asymmetric Telegraph Processes

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 formulae 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 moments is analysed under nonstandard Kac's scaling.

Tunneling as a stochastic process: a path-integral model for microwave experiments

Delay time results obtained in microwave experiments at frequencies above and below the cutoff frequency of different waveguide sections are interpreted on the basis of wave propagation in the presence of dissipative effects. Kac's original suggestion was the starting point for the formulation of a stochastic model, which has now been substantially improved, also in relation to the transition-elements theory of Feynman-Hibbs. In this way, an approach to the problem is provided, which is completely distinct from the ones formulated elsewhere.