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

Levy statistical fluctuations from a Random Amplifying Medium

We report the studies of emission from a novel random amplifying medium that we term a ``Levy Laser'' due to the non-Gaussian statistical nature of its emission over the ensemble of random realizations. It is observed that the amplification is dominated by certain improbable events that are ``larger than rare'', which give the intensity statistics a Levy like ``fat tail''. This, to the best of our knowledge, provides the first experimental realization of Levy flight in optics in a random amplifying medium.Comment: 22 pages, 14 figures (postscript format

Cramer's estimate for the exponential functional of a Levy process

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{\xi_s} ds$ associated to a Levy process $(\xi_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process $\xi$, the main one being Cramer's condition, that asserts the existence of a real $\chi >0$ such that ${\Bbb E}(e^{\chi \xi_1})=1$. Then there exists $C>0$ satisfying, when $t \to +\infty$ : $${\Bbb P} (A_{\infty}> t) \sim C t^{-\chi} \quad .$$ This result can be applied for example to the process $\xi_t = at - S_{\alpha}(t)$ where $S_{\alpha}$ stands for the stable subordinator of index $\alpha$ ($0 < \alpha < 1$), and $a$ is a positive real (we have then $\chi=a^{1/(\alpha -1)}$).Comment: 12 page

Martingale-valued measures, Ornstein-Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space

We investigate a class of Hilbert space valued martingale-valued measures whose covariance structure is determined by a trace class positive operator valued measure. The paradigm example is the martingale part of a Levy process. We develop both weak and strong stochastic integration with respect to such martingale-valued measures. As an application, we investigate the stochastic convolution of a C0-semigroup with a Levy process and the associated Ornstein-Uhlenbeck process. We give an in¯nite dimensional generalisation of the concept of operator self-decomposability and conditions for random variables of this type to be embedded into a stationary Ornstein-Uhlenbeck process

Exact solution of a Levy walk model for anomalous heat transport

The Levy walk model is studied in the context of the anomalous heat conduction of one dimensional systems. In this model the heat carriers execute Levy-walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations and the temperature profile of the Levy-walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is non-locally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size dependent cut-off time is necessary for the Levy walk model to behave as mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes.Comment: 5 pages, 2 figure

Ontology-Based Data Access and Integration

An ontology-based data integration (OBDI) system is an information management system consisting of three components: an ontology, a set of data sources, and the mapping between the two. The ontology is a conceptual, formal description of the domain of interest to a given organization (or a community of users), expressed in terms of relevant concepts, attributes of concepts, relationships between concepts, and logical assertions characterizing the domain knowledge. The data sources are the repositories accessible by the organization where data concerning the domain are stored. In the general case, such repositories are numerous, heterogeneous, each one managed and maintained independently from the others. The mapping is a precise specification of the correspondence between the data contained in the data sources and the elements of the ontology. The main purpose of an OBDI system is to allow information consumers to query the data using the elements in the ontology as predicates. In the special case where the organization manages a single data source, the term ontology-based data access (ODBA) system is used