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.

On the Wick product and Wong-Zakai approximations.

This thesis is a compilation of 6 papers that the author has written together with Alberto Lanconelli (chapters 3, 5 and 8) and Hyun-Jung Kim (ch 7). The logic thread that link all these chapters together is the interest to analyze and approximate the solutions of certain stochastic differential equations using the so called Wick product as the basic tool. In the first chapter we present arguably the most important achievement of this thesis; namely the generalization to multiple dimensions of a Wick-Wong-Zakai approximation theorem proposed by Hu and Oksendal. By exploiting the relationship between the Wick product and the Malliavin derivative we propose an original reduction method which allows us to approximate semi-linear systems of stochastic differential equations of the Itô type. Furthermore in chapter 4 we present a non-trivial extension of the aforementioned results to the case in which the system of stochastic differential equations are driven by a multi-dimensional fraction Brownian motion with Hurst parameter bigger than 1/2. In chapter 5 we employ our approach and present a “short time” approximation for the solution of the Zakai equation from non-linear filtering theory and provide an estimation of the speed of convergence. In chapters 6 and 7 we study some properties of the unique mild solution for the Stochastic Heat Equation driven by spatial white noise of the Wick-Skorohod type. In particular by means of our reduction method we obtain an alternative derivation of the Feynman-Kac representation for the solution, we find its optimal Hölder regularity in time and space and present a Feynman-Kac-type closed form for its spatial derivative. Chapter 8 treats a somewhat different topic; in particular we investigate some probabilistic aspects of the unique global strong solution of a two dimensional system of semi-linear stochastic differential equations describing a predator-prey model perturbed by Gaussian noise.

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity.

Model predictive control in thermal management of multiprocessor systems-on-chip

MultiProcessor Systems-on-Chip (MPSoC) are the core of nowadays and next generation computing platforms. Their relevance in the global market continuously increase, occupying an important role both in everydaylife products (e.g. smartphones, tablets, laptops, cars) and in strategical market sectors as aviation, defense, robotics, medicine. Despite of the incredible performance improvements in the recent years processors manufacturers have had to deal with issues, commonly called “Walls”, that have hindered the processors development. After the famous “Power Wall”, that limited the maximum frequency of a single core and marked the birth of the modern multiprocessors system-on-chip, the “Thermal Wall” and the “Utilization Wall” are the actual key limiter for performance improvements. The former concerns the damaging effects of the high temperature on the chip caused by the large power densities dissipation, whereas the second refers to the impossibility of fully exploiting the computing power of the processor due to the limitations on power and temperature budgets. In this thesis we faced these challenges by developing efficient and reliable solutions able to maximize performance while limiting the maximum temperature below a fixed critical threshold and saving energy. This has been possible by exploiting the Model Predictive Controller (MPC) paradigm that solves an optimization problem subject to constraints in order to find the optimal control decisions for the future interval. A fully-distributedMPC-based thermal controller with a far lower complexity respect to a centralized one has been developed. The control feasibility and interesting properties for the simplification of the control design has been proved by studying a partial differential equation thermal model. Finally, the controller has been efficiently included in more complex control schemes able to minimize energy consumption and deal with mixed-criticalities tasks

Potential Analysis for Hypoelliptic Second Order PDEs with Nonnegative Characteristic Form

In this Thesis we consider a class of second order partial differential operators with non-negative characteristic form and with smooth coefficients. Main assumptions on the relevant operators are hypoellipticity and existence of a well-behaved global fundamental solution. We first make a deep analysis of the L-Green function for arbitrary open sets and of its applications to the Representation Theorems of Riesz-type for L-subharmonic and L-superharmonic functions. Then, we prove an Inverse Mean value Theorem characterizing the superlevel sets of the fundamental solution by means of L-harmonic functions. Furthermore, we establish a Lebesgue-type result showing the role of the mean-integal operator in solving the homogeneus Dirichlet problem related to L in the Perron-Wiener sense. Finally, we compare Perron-Wiener and weak variational solutions of the homogeneous Dirichlet problem, under specific hypothesis on the boundary datum.

Distributed Large-scale Mixed-Integer Optimization with Application to Energy and Multi-robot Networks

Several decision and control tasks in cyber-physical networks can be formulated as large- scale optimization problems with coupling constraints. In these "constraint-coupled" problems, each agent is associated to a local decision variable, subject to individual constraints. This thesis explores the use of primal decomposition techniques to develop tailored distributed algorithms for this challenging set-up over graphs. We first develop a distributed scheme for convex problems over random time-varying graphs with non-uniform edge probabilities. The approach is then extended to unknown cost functions estimated online. Subsequently, we consider Mixed-Integer Linear Programs (MILPs), which are of great interest in smart grid control and cooperative robotics. We propose a distributed methodological framework to compute a feasible solution to the original MILP, with guaranteed suboptimality bounds, and extend it to general nonconvex problems. Monte Carlo simulations highlight that the approach represents a substantial breakthrough with respect to the state of the art, thus representing a valuable solution for new toolboxes addressing large-scale MILPs. We then propose a distributed Benders decomposition algorithm for asynchronous unreliable networks. The framework has been then used as starting point to develop distributed methodologies for a microgrid optimal control scenario. We develop an ad-hoc distributed strategy for a stochastic set-up with renewable energy sources, and show a case study with samples generated using Generative Adversarial Networks (GANs). We then introduce a software toolbox named ChoiRbot, based on the novel Robot Operating System 2, and show how it facilitates simulations and experiments in distributed multi-robot scenarios. Finally, we consider a Pickup-and-Delivery Vehicle Routing Problem for which we design a distributed method inspired to the approach of general MILPs, and show the efficacy through simulations and experiments in ChoiRbot with ground and aerial robots.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Manifestazioni Cranio-facciali delle Neurofibromatosi

The term neurofibromatosis (NF) subsumes at least seven different genetic disorders associated by the presence of neurofibromas located in the skin, oral cavity, visceral and skeletal level. As NF1 (Von Recklinghausen disease), one of the most common genetic diseases, can have oral manifestations, dentists have to be aware about pathognomonic features. The thesis’ target is the literature’s review on the NF1 manifestations either systemic or cefalic area and these features’ research in a specimen of 30 patients NF1 affected. NF1 is manifested in the cefalic area locating either in the jaws (isolated neurofibromas, ipoplasia or bone structures absence) or soft tissues (fibromas and neurofibromas located in: cheeck, lips, oral mucosa, tongue, mouth’s floor, gingiva and palate). Frequently, NF1 patients are affected by dental anomalies of position, number and eruption, that determinates the possibility of orthopaedic-orthodontic problems. An increased prevalence of the caries risk and a possible pulpar involvement of neurofibromas is reported. Clinical and radiographical typical signs of the disease and specific indications for the differential diagnosis with other oral pathologies are described (cysts and odontogenic tumors, periapical lesions of endodontic origin and severe parodontitis). The importance of screening programs and periodical follow-ups (biannual dental visits from the age of four years, annual X-ray checks from the age of six) is supported by the high frequency of manifestations at hard and soft tissues level of the cefalic area and by the documented risk of malignant transformation.

L'Amiloidosi transtiretino correlata ereditaria: profilo clinico, fisiopatologico e storia naturale

Background. Hereditary transthyretin (TTR)-related amyloidosis (ATTR) is mainly considered a neurologic disease. We assessed the phenotypic and genotypic spectrum of ATTR in a non-endemic, Caucasian area and evaluated prevalence, genetic background and disease profile of cases with an exclusively cardiac phenotype, highlighting possible hints for the differential diagnosis with hypertrophic cardiomyopathy (HCM) and senile systemic amyloidosis (SSA) Methods and Results. In this Italian multicenter study, 186 patients with ATTR were characterized at presentation. Thirty patients with SSA and 30 age-gender matched HCM patients were used for comparison. Phenotype was classified as: exclusively cardiac (n= 31, 17%), exclusively neurologic (n= 46, 25%), mixed cardiac/neurologic (n=109, 58%). Among the 8 different mutations responsible for an exclusively cardiac phenotype, Ile68Leu was the most frequent (23/31). Five patients with an exclusively cardiac phenotype developed mild abnormalities at neurological examination but no symptoms during a 36 [14−50] month follow-up. Exclusively cardiac phenotype was characterized by male gender, age > 65 years, heart failure symptoms, concentric left ventricular (LV) “hypertrophy” and moderately depressed LV ejection fraction. This profile was similar to SSA but relatively distinct from HCM. Compared to patients with a mixed phenotype, patients with a exclusively cardiac phenotype showed a more pronounced cardiac involvement on both echocardiogram and ECG. Conclusion. A clinically relevant subset of Caucasian ATTR patients present with an exclusively cardiac phenotype, mimicking HCM or SSA. Echocardiographic and ECG findings are useful to differentiate ATTR from HCM but not from SSA. The role of liver transplantation in these patients is questionable.

Diffusive models and chaos indicators for non-linear betatron motion in circular hadron accelerators

Understanding the complex dynamics of beam-halo formation and evolution in circular particle accelerators is crucial for the design of current and future rings, particularly those utilizing superconducting magnets such as the CERN Large Hadron Collider (LHC), its luminosity upgrade HL-LHC, and the proposed Future Circular Hadron Collider (FCC-hh). A recent diffusive framework, which describes the evolution of the beam distribution by means of a Fokker-Planck equation, with diffusion coefficient derived from the Nekhoroshev theorem, has been proposed to describe the long-term behaviour of beam dynamics and particle losses. In this thesis, we discuss the theoretical foundations of this framework, and propose the implementation of an original measurement protocol based on collimator scans in view of measuring the Nekhoroshev-like diffusive coefficient by means of beam loss data. The available LHC collimator scan data, unfortunately collected without the proposed measurement protocol, have been successfully analysed using the proposed framework. This approach is also applied to datasets from detailed measurements of the impact on the beam losses of so-called long-range beam-beam compensators also at the LHC. Furthermore, dynamic indicators have been studied as a tool for exploring the phase-space properties of realistic accelerator lattices in single-particle tracking simulations. By first examining the classification performance of known and new indicators in detecting the chaotic character of initial conditions for a modulated Hénon map and then applying this knowledge to study the properties of realistic accelerator lattices, we tried to identify a connection between the presence of chaotic regions in the phase space and Nekhoroshev-like diffusive behaviour, providing new tools to the accelerator physics community.

Spectral asymptotic properties of semi-regular non-commutative harmonic oscillators

The study carried out in this thesis is devoted to spectral analysis of systems of PDEs related also with quantum physics models. Namely, the research deals with classes of systems that contain certain quantum optics models such as Jaynes-Cummings, Rabi and their generalizations that describe light-matter interaction. First we investigate the spectral Weyl asymptotics for a class of semiregular systems, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch. Actually, the asymptotics by Doll, Gannot and Wunsch is more precise (that is why we call it refined) than the classical result by Helffer and Robert, but deals with a less general class of systems, since the authors make an hypothesis on the measure of the subset of the unit sphere on which the tangential derivatives of the X-Ray transform of the semiprincipal symbol vanish to infinity order. Abstract Next, we give a meromorphic continuation of the spectral zeta function for semiregular differential systems with polynomial coefficients, generalizing the results by Ichinose and Wakayama and Parmeggiani. Finally, we state and prove a quasi-clustering result for a class of systems including the aforementioned quantum optics models and we conclude the thesis by showing a Weyl law result for the Rabi model and its generalizations.