Telegraph Process Research Articles

Quantum walks of two interacting particles on percolation graphs

We address the dynamics of two indistinguishable interacting particles moving on a dynamical percolation graph, i.e., a graph where the edges are independent random telegraph processes whose values jump between 0 and 1, thus mimicking percolation. The interplay between the particle interaction strength, initial state and the percolation rate determine different dynamical regimes for the walkers. We show that, whenever the walkers are initially localised within the interaction range, fast noise enhances the particle spread compared to the noiseless case.

Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation

Abstract In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part I basic theory

This article introduces the notion of generalized Poisson–Kac (GPK) processes which generalize the class of ‘telegrapher’s noise dynamics’ introduced by Kac (1974 Rocky Mount. J. Math. 4 497) in 1974, using Poissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/long-distance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III (Giona et al 2016a J. Phys. A: Math. Theor., 2016b J. Phys. A: Math. Theor.).

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part II Irreversibility, norms and entropies

In this second part, we analyze the dissipation properties of generalized Poisson–Kac (GPK) processes, considering the decay of suitable L2-norms and the definition of entropy functions. In both cases, consistent energy dissipation and entropy functions depend on the whole system of primitive statistical variables, the partial probability density functions , while the corresponding energy dissipation and entropy functions based on the overall probability density do not satisfy monotonicity requirements as a function of time. These results provide new insights on the theory of Markov operators associated with irreversible stochastic dynamics. Examples from chaotic advection (standard map coupled to stochastic GPK processes) illustrate this phenomenon. Some complementary physical issues are also addressed: the ergodicity breaking in the presence of attractive potentials, and the use of GPK perturbations to mollify stochastic field equations.

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

This third part extends the theory of Generalized Poisson–Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker–Planck–Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac’s program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

Mean first passage times for piecewise deterministic Markov processes and the effects of critical points

In this paper, we use probabilistic methods to determine the mean first passage time (MFPT) for a two-state piecewise deterministic Markov process (PDMP), also known as a dichotomous noise process, to escape from a finite interval. In particular, we consider the case where the set of functions generating the piecewise deterministic dynamics have one or more critical points. In order to solve this type of problem, we partition the domain into a set of subintervals that contain no critical points and impose conditions at the critical points separating these regions. Our analysis exploits the fact that a PDMP satisfies the strong Markov property. We prove that in the absence of common critical points, the MFPT is finite. Through specific examples, we also explore how the MFPT depends on the number of critical points and prove that the MFPT can be infinite if there are common critical points.

Telegraph Process with Elastic Boundary at the Origin

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards, with probability 1−α. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.

Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation

A Brownian motion whose infinitesimal variance alternates according to a telegraph process is considered. This stochastic process can be employed to model a variety of real-word situations, such as animal movement in ecology and stochastic volatility in mathematical finance. The main goal is to develop an estimation procedure for the underlying model parameters when the process is observed at discrete, possibly irregularly spaced time points. The sequence of observations is not Markov, but the sequence of the state of the telegraph process, if observed, is Markov. The observed sequence is therefore from a hidden Markov model. Likelihood inference is developed via dynamic programming, and is demonstrated to have much higher efficiency than the composite likelihood approach that was applied in an earlier work. The model is applied to model the movement of a mountain lion.

“I don't know if I would report them”: Student-athletes' thoughts, feelings and anticipated behaviours on blowing the whistle on doping in sport

Revisions to global anti-doping policy and growing evidence of systematic doping in sport means athletes and athlete support personnel are increasingly encouraged to ‘blow the whistle’ on doping. Yet, individuals' thoughts, feelings, and anticipated behaviours in reporting wrongdoing of this kind are unknown, hindering its promotion. To inform current anti-doping efforts, this study explored student-athletes' anticipated behaviours relative to blowing the whistle on performance enhancing drug (PED) use and their underpinning attitudes. DesignQualitative methodology. MethodSemi-structured interviews were conducted with 28 track and field university student-athletes from the UK (N = 14) and US (N = 14). Transcripts were analysed using thematic analysis. ResultsAddressing doping presents a true moral dilemma and is not a dichotomous process whereby athletes either report doping or do nothing. Instead, four options for addressing others' PED use emerged: (1) confront PED user directly, (2) report to ‘someone’, (3) report to anti-doping ‘authorities’, or (4) ignore the behaviour. Underpinned by relational concerns and empathy, direct confrontation was participants' preferred approach to addressing doping. ConclusionStudent-athletes are reluctant to blow the whistle on doping so the currently promoted method for reporting wrongdoing may be met with resistance. Instead, student-athletes indicate a willingness to personally confront PED users, which has the potential to protect both the doping athlete and whistleblower, while simultaneously reducing the presence of PEDs in sport. Thus, these findings serve to stimulate debate and discussion within anti-doping efforts regarding the possibility of confrontation being encouraged as an effective deterrent to sport doping.

Stochastic Liouville equation for particles driven by dichotomous environmental noise.

We analyze the stochastic dynamics of a large population of noninteracting particles driven by a global environmental input in the form of a dichotomous Markov noise process (DMNP). The population density of particle states evolves according to a stochastic Liouville equation with respect to different realizations of the DMNP. We then exploit the connection with previous work on diffusion in randomly switching environments, in order to derive moment equations for the distribution of solutions to the stochastic Liouville equation. We illustrate the theory by considering two simple examples of dichotomous flows, a velocity jump process and a two-state gene regulatory network. In both cases we show how the global environmental input induces statistical correlations between different realizations of the population density.

Variational principles and Lagrangian functions for stochastic processes and their dissipative statistical descriptions

A variational Lagrangian formulation for stochastic processes and for the evolution equations of the associated probability density functions is developed. Particular attention is dedicated to Poisson–Kac processes possessing finite propagation velocity. The variational formulation in terms of Lagrangian and Hamiltonian densities permits to address different forms of “reversibility” characterizing these processes with respect to processes driven by nowhere differentiable Wiener fluctuations, and associated with the concepts of dynamic and statistical reversibility. The latter property, i.e. the statistical reversibility, implies that the extended Markov operator associated with Poisson–Kac and Generalized Poisson–Kac processes forms a group, continuously parametrized with respect to time.

Rural Migration in the Developing Economy and its Reflection on the Urban Structure (with Reference to Kurdistan Region)

Indeed ,a lot of common lines bringing different experiences of the developing countries (in respect to rural migration) that are closed to each other, however each experience has its special characteristics which are not necessarily congruent to other experiences as has really been taken place during 1980s in Kurdistan region. To deal with practical aspects ofurbanization in Kurdistan region, a simple questionnaire conducted in two urban areas; Sulaimany (city) and Koya (town). The questions were immediately related to the prospected causes and consequences of migration and urban localization. The research directed to test (qualitatively with reference to the questionnaire results) a principal assumption ;( aversion of migration and urban attractions, both of them resulted in a dichotomous process of settlement). In so far this paper concentrated on the following subjects; One – Rural centrifugal forces of migration. Two –Governmental Obligatory and Urban centripetal forcesof migration. Three- Urbanization and immigration consequences. Four – Towards a sound urbanization in the developing economy. Lastly, the paper concluded that after a long and sluggish rural migration in Iraq during 1950s, 1960s and 1970s governmental obligatory forces in the second half of 1980s represented the most decisive and profoundly determinants of the rural migration in the IKRand this , as a basic consequence, entails a chronic restless to reside between the village and the city.

Photoinduced diffusion molecular transport

We consider a Brownian photomotor, namely, the directed motion of a nanoparticle in an asymmetric periodic potential under the action of periodic rectangular resonant laser pulses which cause charge redistribution in the particle. Based on the kinetics for the photoinduced electron redistribution between two or three energy levels of the particle, the time dependence of its potential energy is derived and the average directed velocity is calculated in the high-temperature approximation (when the spatial amplitude of potential energy fluctuations is small relative to the thermal energy). The thus developed theory of photoinduced molecular transport appears applicable not only to conventional dichotomous Brownian motors (with only two possible potential profiles) but also to a much wider variety of molecular nanomachines. The distinction between the realistic time dependence of the potential energy and that for a dichotomous process (a step function) is represented in terms of relaxation times (they can differ on the time intervals of the dichotomous process). As shown, a Brownian photomotor has the maximum average directed velocity at (i) large laser pulse intensities (resulting in short relaxation times on laser-on intervals) and (ii) excited state lifetimes long enough to permit efficient photoexcitation but still much shorter than laser-off intervals. A Brownian photomotor with optimized parameters is exemplified by a cylindrically shaped semiconductor nanocluster which moves directly along a polar substrate due to periodically photoinduced dipole moment (caused by the repetitive excited electron transitions to a non-resonant level of the nanocylinder surface impurity).

VIII. Captured in terminology: Sex, sex categories, and sex differences

Swayed by the clear distinction between male and female genitalia, the question of how far these categories extend into human biology has attracted humans for centuries. This question is sometimes being framed as whether the effects of sex are restricted to the genital organs or penetrate the entire organism. Here I argue that the two questions are not equivalent and that whereas the answer to the question, how far sex penetrates the body, is – deep down to the level of every cell, the answer to the question, how far the categories, “male” and “female”, do, is – probably nowhere beyond the genitals. That the two questions are often used interchangeably reflects the prevailing conceptualization of sex as a dichotomous system or process that exerts profound effects on other systems (e.g., the brain), leading to sexual dimorphism (i.e., two forms, male versus female) also of these systems. Here I discuss the question of whether the effects of sex result in dimorphic systems, focusing on the case of sex effects on the brain. I show that although there are sex/gender differences in brain and behavior, humans and human brains are comprised of highly variable ‘mosaics’ of features, some more prevalent in females, others more prevalent in males.

Long time behavior of telegraph processes under convex potentials

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift towards the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.

A consistency estimate for Kac’s model of elastic collisions in a dilute gas

An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For suitable initial data, with high probability, the two processes agree to within a tolerance of order $N^{-1/d}$, where $N$ is the smaller particle number and $d$ is the dimension, provided that $d\ge3$. From this estimate we can deduce that the spatially homogeneous Boltzmann equation is well posed in a class of measure-valued processes and provides a good approximation to the Kac process when the number of particles is large. We also prove in an appendix a basic lemma on the total variation of time-integrals of time-dependent signed measures.

Generalized Poisson–Kac Processes: Basic Properties and Implications in Extended Thermodynamics and Transport

Abstract We introduce a new class of stochastic processes in R n , $${{\rm{\mathbb R}}^n}, $$ referred to as generalized Poisson–Kac (GPK) processes, that generalizes the Poisson–Kac telegrapher’s random motion in higher dimensions. These stochastic processes possess finite propagation velocity, almost everywhere smooth trajectories, and converge in the Kac limit to Brownian motion. GPK processes are defined by coupling the selection of a bounded velocity vector from a family of N distinct ones with a Markovian dynamics controlling probabilistically this selection. This model can be used as a probabilistic tool for a stochastically consistent formulation of extended thermodynamic theories far from equilibrium.

Energetics of Poisson–Kac Stochastic Processes Possessing Finite Propagation Velocity

Abstract A local fluctuation–dissipation theorem for the power delivered by a stochastic forcing is derived for Ornstein–Uhlenbeck processes driven by smooth, i. e. almost everywhere (a. e.)-differentiable stochastic perturbations (Poisson–Kac processes). An analytic expression for the probability density function of the fluctuational power is obtained in the large time limit. As these processes converge, in the Kac limit, toward classical Langevin equations driven by Wiener processes, a coarse-grained analysis of the statistical properties of the fluctuational work is developed.

Large deviations for some non-standard telegraph processes

We prove large deviation principles for three non-standard telegraph processes. The first one is a damped model with velocity driven by Bernoulli trials studied in Crimaldi et al. (2013), and we obtain the same rate function obtained in De Gregorio and Macci (2014) for another damped telegraph process. The other telegraph processes are non-damped models and we assume suitable hypotheses: in a case the holding times have a general super-exponential distribution, in another case the change-of-direction number process satisfies a large deviation principle.

The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W(t)} have infinitely many zero-crossings in the interval (0, δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of intersections of a level or the number of level-crossings for the telegraph process. Passing to the limit under the Kac condition, we also obtain an estimate of the level-crossings for the Wiener process.