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.

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.

Models for control of highly infectious diseases on local, small-world, and scale-free networks are considered, with only partial information accessible about the status of individuals and their connections. We consider a case when individuals can be infectious before showing symptoms and thus before detection. For small to moderately severe incidence of infection with a small number of nonlocal links, it is possible to control disease spread by using purely local methods applied in a neighborhood centered around a detected infectious individual. There exists an optimal radius for such a control neighborhood leading to the lowest severity of the epidemic in terms of economic costs associated with disease and treatment. The efficiency of a local control strategy is very sensitive to the choice of the radius. Below the optimal radius, the local strategy is unsuccessful; the disease spreads throughout the system, necessitating treatment of the whole population. At the other extreme, a strategy involving a neighborhood that is too large controls the disease but is wasteful of resources. It is not possible to stop an epidemic on scale-free networks by preventive actions, unless a large proportion of the population is treated.

We consider the motion of an overdamped particle in a periodic potential lacking spatial symmetry under the influence of symmetric Lévy noise, being a minimal setup for a "Lévy ratchet." Due to the non-thermal character of the Lévy noise, the particle exhibits a motion with a preferred direction even in the absence of whatever additional time-dependent forces. The examination of the Lévy ratchet has to be based on the characteristics of directionality which are different from typically used measures like mean current and the dispersion of particles' positions, since these get inappropriate when the moments of the noise diverge. To overcome this problem, we discuss robust measures of directionality of transport like the position of the median of the particles displacements' distribution characterizing the group velocity, and the interquantile distance giving the measure of the distributions' width. Moreover, we analyze the behavior of splitting probabilities for leaving an interval of a given length unveiling qualitative differences between the noises with Lévy indices below and above unity. Finally, we inspect the problem of the first escape from an interval of given length revealing independence of exit times on the structure of the potential.

Commonly, normal diffusive behavior is characterized by a linear dependence of the second central moment on time, {x2(t) proportional t, while anomalous behavior is expected to show a different time dependence, x2(t) proportional t{delta} with delta<1 for subdiffusive and delta>1 for superdiffusive motions. Here we explore in details the fact that this kind of qualification, if applied straightforwardly, may be misleading: there are anomalous transport motions revealing perfectly "normal" diffusive character (x2(t) proportional t) yet being non-Markov and non-Gaussian in nature. We use recently developed framework of Monte Carlo simulations which incorporates anomalous diffusion statistics in time and space and creates trajectories of such an extended random walk. For special choice of stability indices describing statistics of waiting times and jump lengths, the ensemble analysis of anomalous diffusion is shown to hide temporal memory effects which can be properly detected only by examination of formal criteria of Markovianity (fulfillment of the Chapman-Kolmogorov equation).

We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness β of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry β causes typically a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index α which characterizes the jump length distribution of Lévy noise, with a marked discontinuity occurring at α = 1. We find that the typical factor of "two" that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-tobarrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior. Distinct non-exponential deviations occur, however, for low barrier heights.

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 not-fully equilibrated bath. Our attention focuses on the effect of the stable noises on the mean escape time and on the phenomenon of resonant activation (RA). Possible physical interpretation of the occurrence of Lévy noises in the system of interest is discussed and the connection between the Tsallis statistics and the Fractional Fokker Planck Equation is addressed.PACS numbers: 05.10.-a, 05.90+m, 02.50.-r, 82.20.-w

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.