12 research outputs found
Levy--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 L\'evy 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 formulae when the stability index approaches 2, recovering
in this limit the standard results for the Fokker-Planck dynamics driven by
Gaussian white noise.Comment: 9 pages, 13 figure
Transport in a Levy ratchet: Group velocity and distribution spread
We consider the motion of an overdamped particle in a periodic potential
lacking spatial symmetry under the influence of symmetric L\'evy noise, being a
minimal setup for a ``L\'evy ratchet.'' Due to the non-thermal character of the
L\'evy noise, the particle exhibits a motion with a preferred direction even in
the absence of whatever additional time-dependent forces. The examination of
the L\'evy 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 L\'evy 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.Comment: 9 pages, 12 figure
Anomalous diffusion and generalized Sparre-Andersen scaling
We are discussing long-time, scaling limit for the anomalous diffusion
composed of the subordinated L\'evy-Wiener process. The limiting anomalous
diffusion is in general non-Markov, even in the regime, where ensemble averages
of a mean-square displacement or quantiles representing the group spread of the
distribution follow the scaling characteristic for an ordinary stochastic
diffusion. To discriminate between truly memory-less process and the non-Markov
one, we are analyzing deviation of the survival probability from the (standard)
Sparre-Andersen scaling.Comment: 5 pages, 3 figure
Stationary states in Langevin dynamics under asymmetric L\'evy noises
Properties of systems driven by white non-Gaussian noises can be very
different from these systems driven by the white Gaussian noise. We investigate
stationary probability densities for systems driven by -stable L\'evy
type noises, which provide natural extension to the Gaussian noise having
however a new property mainly 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
-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.Comment: 9 pages, 9 figures, 3 table
Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis
A standard approach to analysis of noise-induced effects in stochastic
dynamics assumes a Gaussian character of the noise term describing interaction
of the analyzed system with its complex surroundings. An additional assumption
about the existence of timescale separation between the dynamics of the
measured observable and the typical timescale of the noise allows external
fluctuations to be modeled as temporally uncorrelated and therefore white.
However, in many natural phenomena the assumptions concerning the
abovementioned properties of "Gaussianity" and "whiteness" of the noise can be
violated. In this context, in contrast to the spatiotemporal coupling
characterizing general forms of non-Markovian or semi-Markovian L\'evy walks,
so called L\'evy flights correspond to the class of Markov processes which
still can be interpreted as white, but distributed according to a more general,
infinitely divisible, stable and non-Gaussian law. L\'evy noise-driven
non-equilibrium systems are known to manifest interesting physical properties
and have been addressed in various scenarios of physical transport exhibiting a
superdiffusive behavior. Here we present a brief overview of our recent
investigations aimed to understand features of stochastic dynamics under the
influence of L\'evy white noise perturbations. We find that the archetypal
phenomena of noise-induced ordering are robust and can be detected also in
systems driven by non-Gaussian, heavy-tailed fluctuations with infinite
variance.Comment: 7 pages, 8 figure
Escape driven by -stable white noises
We explore the archetype problem of an escape dynamics occurring in a
symmetric double well potential when the Brownian particle is driven by {\it
white L\'evy 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
L\'evy white noise parameters. Depending on the value of the skewness
of the L\'evy noise, the escape can either become enhanced or suppressed: a
negative asymmetry causes typically a decrease for the escape rate
while the rate itself depicts a non-monotonic behavior as a function of the
stability index which characterizes the jump length distribution of
L\'evy noise, with a marked discontinuity occurring at . 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-to-barrier-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.Comment: 8 pages, 8 figure
Understanding disease control: influence of epidemiological and economic factors
We present a local spread model of disease transmission on a regular network
and compare different control options ranging from treating the whole
population to local control in a well-defined neighborhood of an infectious
individual. Comparison is based on a total cost of epidemic, including cost of
palliative treatment of ill individuals and preventive cost aimed at
vaccination or culling of susceptible individuals. Disease is characterized by
pre- symptomatic phase which makes detection and control difficult. Three
general strategies emerge, global preventive treatment, local treatment within
a neighborhood of certain size and only palliative treatment with no
prevention. The choice between the strategies depends on relative costs of
palliative and preventive treatment. The details of the local strategy and in
particular the size of the optimal treatment neighborhood weakly depends on
disease infectivity but strongly depends on other epidemiological factors. The
required extend of prevention is proportional to the size of the infection
neighborhood, but this relationship depends on time till detection and time
till treatment in a non-nonlinear (power) law. In addition, we show that the
optimal size of control neighborhood is highly sensitive to the relative cost,
particularly for inefficient detection and control application. These results
have important consequences for design of prevention strategies aiming at
emerging diseases for which parameters are not known in advance