Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND
NANOSCIENCES

Thesis title: Seven Little pieces of Gentleness

In this dissertation the concept of Gentleness will be defined and analyzed with respect to Stochastic Processes from a physical-mathematical point of view. After a brief introduction just meant to fix the concept of gentleness concerning the field of Stochastic Processes applied to physics, mostly related to the finiteness of the propagation velocity characterizing a stochastic dynamic, and to give some definitions about bounded and unbounded Stochastic Processes, the reader will be able to find 7 little pieces of gentleness containing the basic ideas related to a different paradigm in the application of stochastic processes to physics. Starting by analyzing the idea of gentleness in transport models, we will show how an hyperbolic structure of the balance equations for the probability density functions can be naturally recovered just by enforcing physical consistence to the lattice parameters in the framework of lattice random walk. Throughout the concept of stochastic regularity, strictly connected with the Stochastic Gentleness, it is possible to correctly predict the effective diffusion coefficient in systems of non-interacting particles randomly moving on a two-phases lattice. The middlegame of this thesis will present and analyze in deep the hyperbolic representation of LW statistics in terms of partial density waves parametrized with respect to the transitional age and the direction of propagation, directly relating LWs to other classes of processes possessing finite propagation velocity, such as PoissonKac and Generalized Poisson-Kac processes. Then a proposal to solve the fundamental problem of unbounded random fluctuations is presented by constructing a comprehensive theoretical framework of stochastic processes able to be gentle. To conclude, the last part of this thesis will show 2 recent applications related to the generalization of counting processes through the age formalism of Lévy Walks and the fluctuation-dissipation theorem, underlining how it can be particularly convenient describing hydrodynamic fluctuations by means of Extended Poisson-Kac Processes.