Stochastic Models for Complex Systems

The present project aims at developing mathematical methodologies (both analytic and numerical) in order to describe the behavior and the evolution of a complex system in terms of a stochastic model. Stochastic models are nowadays ubiquitous in science, but sometimes they are build under strong assumptions that may limit their use in applications and/or may weaken their ability to predict the future evolution of a phenomenon. Here, we focus on the study of stochastic systems in order to relax some main assumptions frequently encountered in various applications. These include, among others, the GAUSSIANITY in a finite-dimensional system, the BROWNIAN MOTION in a time-varying context, the LACK-OF-MEMORY PROPERTY of a stochastic process (or its related Markovian assumption). For these purposes, the project will develop novel facets in different types of stochastic models that, under various aspects, interact with each other. To present clear illustrations, the objectives are described according to SIX WORKING PACKAGES (WP).

The description of stochastic systems (with multiple components) is related to an in-depth understanding of the notion of DEPENDENCE, which is nowadays modeled by means of COPULAS. Currently used copula-based models are mainly based on the assumption of CONTINUITY of the marginal distributions (which guarantees the identification of a unique copula to describe the joint probability law). However, in practical applications (and, especially, in environmental ones), random variables may have a discrete component and, hence, observations may contain repeated observations (called ties), which may adversely affect the statistical analysis. In this respect, we aim at: (a) Introducing specific measures, based on the notion of ENTROPY, to quantify possible biases in the modeling process induced by the presence of ties, and to provide suitable general guidelines for a risk estimation; (b) Investigating new association measures and appropriate STOCHASTIC ORDERS for random systems involving a non-continuous random vector. We will also include suitable characterizations of multivariate discrete distributions.

Multivariate stochastic systems have been often used in RELIABILITY THEORY in order to understand the behavior of an engineering system formed by several components. Here, we focus on those systems whose units are linked through multiple types of connections. Specifically, we plan to focus on coherent systems with shared components and/or block-wise structure. Among the different strategies used for the analysis of coherent systems, we are particularly interested in GINI-TYPE INDICES. We plan to extend the stochastic order defined in terms of Gini-type index to more general instances, in order to compare the ageing properties of systems by means of the characteristics of their components. Moreover, we expect to investigate AGEING PROPERTIES of components lifetime that can differ when the other components are working or have already failed. The proposed investigation will receive great benefit from the use of methods and techniques proposed within the framework of the information theory of stochastic systems. Moreover, it will require the development of suitable copula-based models (see WP1) taking into account special interactions among the involved variables.

The exponential curve is the most common basic model to describe growth of populations in ideal conditions. Anyway, such kind of growth does not occur in nature, apart from short time periods. For most living species, indeed, there exists a critical density beyond which the relative population does not find sufficient environmental resources to grow and reproduce. Mathematical models that take into account environmental factors that limit the growth rate of a population are, among others, the logistic model, and other models due to von Bertalanffy, Richards, Smith, Blumberg, Schnute. In this WP, we aim to extend evolution model to the fractional case. The advantage of using Fractional Calculus lies, among other things, in the possibility of introducing more degrees of freedom in the description of certain phenomena, and taking into account the MEMORY of the system. In this setting, we aim to investigate the main properties of evolution models based on fractional derivatives, with special regard to the correction factor, the relative growth rate, the inflection point, and the first-passage-time through a given threshold.

Related to previous (WP3), we will consider more refined evolution models where the phenomena under consideration are subject to random fluctuations governed by increasing/decreasing trends, possibly with jumps. The TELEGRAPH PROCESS is a prototypical example of such situations that describes a random motion with finite velocity on the real line. It is characterized by a probability law governed by a hyperbolic partial differential equation (the telegraph equation) widely used in mathematical physics. Under suitable conditions, it tends to the Brownian motion process. Several generalizations of the telegraph process have been inspired by the need to model physical systems in the presence of a variety of complex conditions, such as the presence of suitable boundaries. Moreover, we propose to investigate multivariate generalizations of the telegraph process, like the Rayleigh-type composition of telegraph processes.

Semi-Markov processes can be considered as the easiest generalization of Markov processes since they are constructed in the same way without assuming the exponential distribution of time intervals, which is crucial to have the Markov property. Hence, the lack of memory property of waiting times may not be satisfied. Heuristically this implies that the evolution after a time s>0 depends not only on the position (as it is for Markov processes) but also on the current amount of time spent by the process in that position. Recent developments showed a strong relationship between integro-differential equations (of fractional type, see WP3) and semi-Markov process, but this connection must be further investigated to be clearly established.

Diffusion processes and Gauss-Markov processes have given relevant benefits in the context of neuronal modeling: among others, we recall the probability density functions of their first passage times (fpt), as solutions of non-singular Volterra equations of the second kind, useful to model the potential of the neuronal membrane. Recently, semi-Markov processes have been considered for neuronal modeling. A particular kind of semi-Markov process (see WP5) can be obtained from Markov processes by suitable time changes, which involves the use of the inverse of a subordinator. Starting from some properties of the Markov process and the variation order of the Lévy exponent of the subordinator, one can also characterize fpt’s density of such process, deriving asymptotic behaviors such as the power-law decay of the related densities.In this WP, we intend to conduct a comparative study between the use of the first and the second aforementioned stochastic processes with a twofold aim: (i) to carry out new theoretical findings involving fpt of semi-Markov processes as counterparts of those valid for Markov processes; (ii) to construct new neuronal models which could include memory effects and multi-scale dynamics.