Related papers: Single integro-differential wave equation for L\'e…
The L\'evy walk process for a lower interval of an excursion times distribution ($\alpha<1$) is discussed. The particle rests between the jumps and the waiting time is position-dependent. Two cases are considered: a rising and diminishing…
This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the…
The standard Levy walk is performed by a particle that moves ballistically between randomly occurring collisions, when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events…
We consider a one dimensional asymmetric random walk whose jumps are identical, independent and drawn from a distribution \phi(\eta) displaying asymmetric power law tails (i.e. \phi(\eta) \sim c/\eta^{\alpha +1} for large positive jumps and…
We investigate a L\'evy-Walk alternating between velocities $\pm v_0$ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic…
Levy walk (LW) process has been used as a simple model for describing anomalous diffusion in which the mean squared displacement of the walker grows non-linearly with time in contrast to the diffusive motion described by simple random walks…
Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in…
The concept of walking wave is introduced from classical relativistic positions. One- and three-dimensional walking waves considered with their wave equations and dispersion equations. It is shown that wave characteristics (de Broglie's and…
L\'evy walk process is one of the most effective models to describe superdiffusion, which underlies some important movement patterns and has been widely observed in the micro and macro dynamics. From the perspective of random walk theory,…
We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random…
The classical relativistic wave equations are presented as partial difference equations in the arena of covariant discrete phase space. These equations are also expressed as difference-differential equations in discrete phase space and…
By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum…
We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal…
We construct a coupling between the random walk composed of L\'evy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used…
A Levy walk is a non-Markovian stochastic process in which the elementary steps of the walker consist of motion with constant speed in randomly chosen directions and for a random period of time. The time of flight is chosen from a…
In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed…
Let $X$ be a L\'evy process with regularly varying L\'evy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed…
We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that…
We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order non-linear conservation law where the flux function includes an integral term. We show that there exist unique…
Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed…