Abstract:
Levy flights, characterized by the microscopic step index f, are for f<2 (the case of rare events) considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f-2 in the short range case and the critical fall-off exponent 2f-2 in the long range case. By means of a dynamic renormalization group analysis based on the momentum shell integration method, we determine flows, fixed point, and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent d=2, two critical dimensions appear in the Levy case. A critical dimension (or fall-off exponent) d=f below which the diffusion coefficient exhibits anomalous scaling behavior, i.e, algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) d=2f-2 below which the force correlations characterized by a non trivial fixed point become relevant. As a general result we find in all cases that the dynamic exponent z, characterizing the mean square displacement, locks onto the Levy index f, independent of dimension and independent of the presence of weak quenched disorder.

Abstract:
The use of reaction-diffusion models rests on the key assumption that the underlying diffusive process is Gaussian. However, a growing number of studies have pointed out the prevalence of anomalous diffusion, and there is a need to understand the dynamics of reactive systems in the presence of this type of non-Gaussian diffusion. Here we present a study of front dynamics in reaction-diffusion systems where anomalous diffusion is due to the presence of asymmetric Levy flights. Our approach consists of replacing the Laplacian diffusion operator by a fractional diffusion operator, whose fundamental solutions are Levy $\alpha$-stable distributions. Numerical simulation of the fractional Fisher-Kolmogorov equation, and analytical arguments show that anomalous diffusion leads to the exponential acceleration of fronts and a universal power law decay, $x^{-\alpha}$, of the tail, where $\alpha$, the index of the Levy distribution, is the order of the fractional derivative.

Abstract:
Levy flights representation is proposed to describe earthquake characteristics like the distribution of waiting times and position of hypocenters in a seismic region. Over 7500 microearthquakes and earthquakes from 1985 to 1994 were analyzed to test that its spatial and temporal distributions are such that can be described by a Levy flight with anomalous diffusion (in this case in a subdiffusive regime). Earthquake behavior is well described through Levy flights and Levy distribution functions such as results show.

Abstract:
We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological L\'{e}vy process with the very same invariant pdf and in the reverse.

Abstract:
We investigate the impact of external periodic potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy index $\mu $. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks.

Abstract:
Let L(t) be a Levy flights process with a stability index \alpha\in(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U'(Z(t-))dt+\sigma(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' \sigma(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. \sigma(t)\approx t^{-\theta} for some \theta>0. We discover two different cooling regimes. If \theta<1/\alpha (slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is concentrated at the potential's local minima. If \theta>1/\alpha (fast cooling) the Levy particle gets trapped in one of the potential wells.

Abstract:
We consider a system of particles undergoing the branching and annihilating reactions A -> (m+1)A and A + A -> 0, with m even. The particles move via long-range Levy flights, where the probability of moving a distance r decays as r^{-d-sigma}. We analyze this system of branching and annihilating Levy flights (BALF) using field theoretic renormalization group techniques close to the upper critical dimension d_c=sigma, with sigma<2. These results are then compared with Monte-Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an active phase occurs at zero branching. However, for sigma bigger than about 3/2 in d=1, the critical branching rate moves smoothly away from zero with increasing sigma, and the transition lies in a different universality class, inaccessible to controlled perturbative expansions. We measure the exponents in both universality classes and examine their behavior as a function of sigma.

Abstract:
On the basis of multivariate Langevin processes we present a realization of Levy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Levy flights. Our procedure is similar to the treatment of the Kramers-Fokker Planck equation in the Smoluchowski limit. The proposed approach forms a feasible way of tackling Levy flights in inhomogeneous media or systems with boundaries what is up to now a challenging problem.

Abstract:
An impact of integration over the paths of the Levy flights on the quantum mechanical kernel has been studied. Analytical expression for a free particle kernel has been obtained in terms of the Fox H-function. A new equation for the kernel of a partical in the box has been found. New general results include the well known quantum formulae for a free particle kernel and particle in box kernel.

Abstract:
Truncated L\'{e}vy flights are random walks in which the arbitrarily large steps of a L\'{e}vy flight are eliminated. Since this makes the variance finite, the central limit theorem applies, and as time increases the probability distribution of the increments becomes Gaussian. Here, truncated L\'{e}vy flights with correlated fluctuations of the variance (heteroskedasticity) are considered. What makes these processes interesting is the fact that the crossover to the Gaussian regime may occur for times considerably larger than for uncorrelated (or no) variance fluctuations. These processes may find direct application in the modeling of some economic time series.