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.

Il dolore oncologico come malattia nella malattia. Una ricerca nazionale sulle donne con tumore al seno

L’ipotesi di fondo su cui si basa l’intero lavoro è che il dolore oncologico debba essere riconosciuto come “malattia nella malattia”: non si può considerare tale dolore mero “sintomo” del cancro ma esperienza totale che coinvolge l’intera persona. Il dolore oncologico è carico di valenze e significati personali, è associato a rappresentazioni sociali e, come ogni malattia, è disease, illness e sickness. Partendo da questo presupposto, la dissertazione si è posta come obiettivo generale quello di studiare il dolore oncologico tra le donne con tumore al seno, le sue componenti sociali, psicologiche, individuali oltre che fisiche; si è voluto inoltre studiare la specificità del vissuto e dei significati associati all’esperienza dolorosa. Il lavoro è articolato in due parti fondamentali, una teorica ed una empirica. La prima presenta un inquadramento dei principali concetti della sociologia della salute riguardanti il dolore. Per quanto riguarda la parte empirica, si è fatto ricorso ad una ricerca mista, fatta di metodi misti e fondata su un approccio metodologico di natura integrativa che si avvale di tecniche quantitative e qualitative. La parte quantitativa si basa su una parte dei dati della ricerca nazionale ESOPO - Epidemiological Study of Pain in Oncology. Dall’intero campione sono state isolate le sole donne con tumore al seno (n=846). Si è proceduto quindi allo studio di tale campione, alle elaborazioni statistiche con il programma SPSS e all’interpretazione dei risultati. Per quanto riguarda la parte qualitativa, invece, è stata condotta un’analisi delle fonti che si è avvalsa di un approccio netnografico: è stata condotta un’osservazione non intrusiva di 12 blog scritti da donne con tumore al seno, con lo scopo di indagare le narrazioni di malattia, i vissuti personali, i significati di dolore e malattia e le loro ripercussioni sulla vita quotidiana.