Single integro-differential wave equation for L\'evy walk

The integro-differential wave equation for the probability density function for a classical one-dimensional L\'evy walk with continuous sample paths has been derived. This equation involves a classical wave operator together with memory integrals describing the spatio-temporal coupling of the L\'evy walk. It is valid for any running time PDF and it does not involve any long-time large-scale approximations. It generalizes the well-known telegraph equation obtained from the persistent random walk. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times.Comment: 5 page

Quantum regime of laser-matter interactions at extreme intensities

A survey of physical parameters and of a ladder of various regimes of laser-matter interactions at extreme intensities is given. Special emphases is made on three selected topics: (i) qualitative derivation of the scalings for probability rates of the basic processes; (ii) self-sustained cascades (which may dominate at the intensity levels attainable with next generation laser facilities); and (iii) possibility of breaking down the Intense Field QED approach for ultrarelativistic electrons and high-energy photons at certain intensity level.Comment: To be published in the Proceedings of the Summer School "Quantum Field Theory at the Limits: from Strong Fields to Heavy Quarks" (18-30 July 2016, BLTP, JINR, Dubna, Russia

Nonlinear subdiffusive fractional equations and aggregation phenomenon

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean field density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.Comment: 10 page

Sub-diffusion in External Potential: Anomalous hiding behind Normal

We propose a model of sub-diffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model. Constant force leads to the transition from non-ergodic sub-diffusion to seemingly ergodic diffusive behavior. However, we show it remains anomalous in a sense that the diffusion coefficient depends on the force and the anomalous exponent. For the quadratic potential we find that the anomalous exponent defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.Comment: 6 pages, 3 figure

An exact renormalization formula for Gaussian exponential sums and applications

In the present paper, we derive a renormalization formula "\`a la Hardy-Littlewood" for the Gaussian exponential sums with an exact formula for the remainder term. We use this formula to describe the typical growth of the Gaussian exponential sums

Stochastic arbitrage return and its implications for option pricing

The purpose of this work is to explore the role that arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility "smile" can also be explained in terms of random arbitrage opportunities.Comment: 14 pages, 3 fiqure

REMOTE-SENSING METHODS OF INDICATOR ESTIMATIONS OF GARDENING TERRITORIES PLACED BY MINING INDUSTRY WASTE

Today across the world there are huge areas that are occupied by badlands left after intensive mining. Breeding dumps, sludge dumps, places of storage of ash and slag often represent a biological desert, which is difficult to remediate. For example, such places are sulfur rock dumps of mines in Donbas as a result of insufficient mineral nutrition and high acidity with sulfur concentration. Such zones show a low rate of self-growth and gardening. Mining wastes that accumulated for many years contain toxic components that are priority sources of environmental pollution