Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Osteogenesi imperfetta e Dentinogenesi imperfetta Tipo I: correlazioni cliniche e istologiche

Osteogenesis imperffecta (OI) is a heterogeneous group of heritable connetive tissue diseases, quantity and/or qualitative defect in type 1 collagen syntesis; sometimes and in some types it can be associated to dentinogenesis imperfecta (DI), a hereditary disorder in dentin formation that comprises a group of autosomal dominant genetic conditions characterized by abnormal dentine structure affecting either the primary or both the primary and secondary dentitions. Aim: the aim of this study was to assess the correlation between OI and DI from both a clinical and histological point of view, clarifying the structural and ultrastructural changes. Eighteen children (&-15 years aged) with diagnosis of OI were examined for dental alterations referable to DI; for each patient, the OI type (I, III, IV) was recorded. Extracted or normally exfolied teeth were subjected to a histological examination.Results: a total of eleven patients had abnormal discolourations referable to DI: five patients were affected by OI type I, three by OI III, and three patients by OI type IV. The discolourations, yellow/brown or oplaescent grey, could not be related to the different types of OI. Histological exam of primary teeth showed severe pathological change in dentin, structured into four diffeent layers. A collagen defect due to odontoblast dysfunction was theorized to be on the base of the histological changes. Conclusions: there is no correlation between the type of OI and the type of discolouration. The underlying dentinal defect seems to be related to an odontoblast dysfunction.

Cooperative Cognitive Wireless Networks over TV White and Gray Spaces

Wireless networks rapidly became a fundamental pillar of everyday activities. Whether at work or elsewhere, people often benefits from always-on connections. This trend is likely to increase, and hence actual technologies struggle to cope with the increase in traffic demand. To this end, Cognitive Wireless Networks have been studied. These networks aim at a better utilization of the spectrum, by understanding the environment in which they operate, and adapt accordingly. In particular recently national regulators opened up consultations on the opportunistic use of the TV bands, which became partially free due to the digital TV switch over. In this work, we focus on the indoor use of of TVWS. Interesting use cases like smart metering and WiFI like connectivity arise, and are studied and compared against state of the art technology. New measurements for TVWS networks will be presented and evaluated, and fundamental characteristics of the signal derived. Then, building on that, a new model of spectrum sharing, which takes into account also the height from the terrain, is presented and evaluated in a real scenario. The principal limits and performance of TVWS operated networks will be studied for two main use cases, namely Machine to Machine communication and for wireless sensor networks, particularly for the smart grid scenario. The outcome is that TVWS are certainly interesting to be studied and deployed, in particular when used as an additional offload for other wireless technologies. Seeing TVWS as the only wireless technology on a device is harder to be seen: the uncertainity in channel availability is the major drawback of opportunistic networks, since depending on the primary network channel allocation might lead in having no channels available for communication. TVWS can be effectively exploited as offloading solutions, and most of the contributions presented in this work proceed in this direction.