Bartłomiej Dybiec

Lévy-Brownian motion on finite intervals : mean first passage time analysis

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.

Modelling control of epidemics spreading by long-range interactions

We have studied the spread of epidemics characterized by a mixture of local and non-local interactions. The infection spreads on a two-dimensional lattice with the fixed nearest neighbour connections. In addition, long-range dynamical links are formed by moving agents (vectors). Vectors perform random walks, with step length distributed according to a thick-tail distribution. Two distributions are considered in this paper, an α-stable distribution describing self-similar vector movement, yet characterized by an infinite variance and an exponential power characterized by a large but finite variance. Such long-range interactions are hard to track and make control of epidemics very difficult. We also allowed for cryptic infection, whereby an infected individual on the lattice can be infectious prior to showing any symptoms of infection or disease. To account for such cryptic spread, we considered a control strategy in which not only detected, i.e. symptomatic, individuals but also all individuals within a certain control neighbourhood are treated upon the detection of disease. We show that it is possible to eradicate the disease by using such purely local control measures, even in the presence of long-range jumps. In particular, we show that the success of local control and the choice of the optimal strategy depend in a non-trivial way on the dispersal patterns of the vectors. By characterizing these patterns using the stability index of the α-stable distribution to change the power-law behaviour or the exponent characterizing the decay of an exponential power distribution, we show that infection can be successfully contained using relatively small control neighbourhoods for two limiting cases for long-distance dispersal and for vectors that are much more limited in their dispersal range.

Resonant activation in the presence of nonequilibrated baths.

We study the generic problem of the escape of a classical particle over a fluctuating barrier under the influence of non-Gaussian noise mimicking the effects of nonequilibrated bath. The model system is described by a Langevin equation with two independent noise sources, one of which stands for the dichotomous process and the other describes external driving by alpha-stable noise. Our attention focuses on the effect of the structure of stable noises on the mean escape time and on the phenomenon of resonant activation. Possible physical interpretation of the occurrence of Lévy noises and the relevance of the model for chemical kinetics is briefly discussed.

Stationary states in Langevin dynamics under asymmetric Lévy noises

Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by alpha-stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of alpha-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.

Stationary states in single-well potentials under symmetric Lévy noises

We discuss the existence of stationary states for subharmonic potentials , c 2 ? ?. Consequently, for harmonic (c = 2) and superharmonic potentials (c > 2) driven by any ?-stable noise, steady states always exist. Stationary states are characterized by probability density functions for having a lighter tail than the noise distribution for superharmonic potentials (c > 2) and a heavier tail than the noise distribution for subharmonic ones. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of the corresponding probability densities.

Lévy ratchet in a weak noise limit: Theory and simulation

Abstract. We study the motion of a particle in a time-independent periodic potential with broken mirror symmetry under action of a Lévy-stable noise (Lévy ratchet). We develop an analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A 39, L237 (2006); Stoch. Proc. Appl. 116, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle’s motion, namely for the splitting probabilities of the first escape from a single well, for the transition probabilities to other wells and for the probability current. We pay particular attention to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Extensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle.

Anomalous diffusion on finite intervals

We study the properties of anomalous diffusion on finite intervals. The process studied due to the presence of trapping events and long jumps is described by a double-fractional (time and space) Fokker–Planck equation. The properties of the overall process are affected not only by long waiting times and long jumps but also by boundaries. Special attention is given to the examination of the survival probability and the first-passage-time density. Using analytical arguments and numerical methods, we show that the asymptotic form of the survival probability is determined by the trapping process. For a special choice of parameters, we compare numerical results with theoretical formulae, demonstrating that numerical solutions constructed by subordination methods reconstruct known analytical results very well. Finally, we show that the power-law distribution of waiting times is responsible for the divergence of the mean first-passage time even for a power-law distribution of jump lengths.

Information spreading and development of cultural centers.

The historical interplay between societies is governed by many factors, including in particular the spreading of languages, religion, and other symbolic traits. Cultural development, in turn, is coupled to the emergence and maintenance of information spreading. Strong centralized cultures exist due to attention from their members, whose faithfulness in turn relies on the supply of information. Here we discuss a culture evolution model on a planar geometry that takes into account aspects of the feedback between information spreading and its maintenance. Features of the model are highlighted by comparing it to cultural spreading in ancient and medieval Europe, where it suggests in particular that long-lived centers should be located in geographically remote regions.

Lévy flights versus Lévy walks in bounded domains

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities is the discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. As a consequence, a well-developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flight and Lévy walk models on bounded domains, examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time, and stationary probability density functions. It is demonstrated that the similarity of the models is affected by the type of boundary conditions and the value of the stability index defining the asymptotics of the jump length distribution.

Escape from the potential well: competition between long jumps and long waiting times.

Within a concept of the fractional diffusion equation and subordination, the paper examines the influence of a competition between long waiting times and long jumps on the escape from the potential well. Applying analytical arguments and numerical methods, we demonstrate that the presence of long waiting times distributed according to a power-law distribution with a diverging mean leads to very general asymptotic properties of the survival probability. The observed survival probability asymptotically decays like a power law whose form is not affected by the value of the exponent characterizing the power law jump length distribution. It is demonstrated that this behavior is typical of and generic for systems exhibiting long waiting times. We also show that the survival probability has a universal character not only asymptotically, but also at small times. Finally, it is indicated which properties of the first passage time density are sensitive to the exact value of the exponent characterizing the jump length distribution.