A planar random motion with an infinite number of directions controlled by the damped wave 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.

Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher's equations-as well as their fractional generalizations-from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations.

Two-dimensional telegraphic processes and their fractional generalizations

We study the planar motion of telegraphic processes. We derive the two-dimensional telegrapher's equation for isotropic and uniform motions starting from a random walk model which is the two-dimensional version of the multistate random walk with a continuum number of states representing the spatial directions. We generalize the isotropic model and the telegrapher's equation to include planar fractional motions. Earlier, we worked with the one-dimensional version [Masoliver, Phys. Rev. E 93, 052107 (2016)PREHBM2470-004510.1103/PhysRevE.93.052107] and derived the three-dimensional version [Masoliver, Phys. Rev. E 96, 022101 (2017)PREHBM2470-004510.1103/PhysRevE.96.022101]. An important lesson is that we cannot obtain the two-dimensional version from the three-dimensional or the one-dimensional one from the two-dimensional result. Each dimension must be approached starting from an appropriate random walk model for that dimension.

Piecewise deterministic processes following two alternating patterns

We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied.

Three-dimensional telegrapher's equation and its fractional generalization

We derive the three-dimensional telegrapher's equation out of a random walk model. The model is a three-dimensional version of the multistate random walk where the number of different states form a continuum representing the spatial directions that the walker can take. We set the general equations and solve them for isotropic and uniform walks which finally allows us to obtain the telegrapher's equation in three dimensions. We generalize the isotropic model and the telegrapher's equation to include fractional anomalous transport in three dimensions.

A Note on Planar Random Motion at Finite Speed

A simple derivation of the explicit form of the transition density of a planar random motion at finite speed, based on some specific properties of the wave propagation on the plane R2, is given.

A four-dimensional random motion at finite speed

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Stochastic velocity motions and processes with random time

The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at time t > 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.