Related papers: Single integro-differential wave equation for L\'e…
We construct an estimator of the L\'evy density of a pure jump L\'evy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the…
The Weierstrass random walk is a paradigmatic Markov chain giving rise to a L\'evy-type superdiffusive behavior. It is well known that Special Relativity prevents the arbitrarily high velocities necessary to establish a superdiffusive…
We introduce a L\'evy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers $\{\xi_i \}$ are independent random variables identically…
We consider a discrete-time random walk where the random increment at time step $t$ depends on the full history of the process. We calculate exactly the mean and variance of the position and discuss its dependence on the initial condition…
We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line…
In this work we present the fundamental ideas of inference over paths, and show how this formalism implies the continuity equation, which is central for the derivation of the main partial differential equations that constitute…
A one-way wave equation is an evolution equation in one of the space directions that describes (approximately) a wave field. The exact wave field is approximated in a high frequency, microlocal sense. Here we derive the pseudodifferential…
We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$,…
We consider a system of interacting particles subjected to Langevin inertial dynamics and derive the governing time-dependent equation for the one-body density. We show that, after suitable truncations of the…
We investigate the relation of the semigroup probability density of an infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first…
Statistics of molecular random walks in a fluid is considered with the help of the Bogolyubov equation for generating functional of distribution functions. An invariance group of solutions to this equation as functions of the fluid density…
An equation describing the irreversible evolution of the local density of a continuous medium without involving any statistical hypotheses and assumptions is derived. The derivation is based on the smoothing of the microscopic dynamic…
The combined Continuous Time Random Walk (CTRW) in position and momentum space is introduced, in the form of two coupled integral equations that describe the evolution of the probability distribution for finding a particle at a certain…
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz…
We show that probability is locally conserved in discrete time quantum walks, corresponding to a particle evolving in discrete space and time. In particular, for a spatial structure represented by an arbitrary directed graph, and any…
In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering…
The density of states for a particle moving in a random potential with a Gaussian correlator is calculated exactly using the functional integral technique. It is achieved by expressing the functional degrees of freedom in terms of the…
This paper is an investigation of the class of real classical Markov processes without a birth process that will generate the Dirac equation in 1+1 dimensions. The Markov process is assumed to evolve in an extra (ordinal) time dimension.…
The Dirac equation can be modelled as a quantum walk, with the quantum walk being: discrete in time and space (i.e. a unitary evolution of the wave-function of a particle on a lattice); homogeneous (i.e. translation-invariant and…
We formulate the first order Fermi acceleration in parallel shock waves in terms of the random walk theory. The formulation is applicable to any value of the shock speed and the particle speed, in particular to the acceleration in…