Random walks in non-homogeneous environments: fluctuation theory, statistical issues and transport properties

Random walks and diffusive particles are one of the most basic and well-studied topics in probability theory, which have been deepened for years both from a purely mathematical point of view and for interesting application possibilities in sciences and beyond. One of the fundamental discoveries, which is a milestone for the birth and development of the field of stochastic processes, is attributed to the biologist Robert Brown in 1827, who observed the random movements performed by minute particles immersed on the surface of water. The first mathematical descriptions of the so called Brownian motion, however, appeared only several years later. In this regard, one should mention the first attempts by Thorvald N. Thiele in 1880 and Louis Bachelier in 1900, although an exhaustive probabilistic theory was not available until Albert Einstein in 1905 and independently Marian Smoluchowski in 1906 provided a physical description of the phenomenon by means of equations accounting for a random number of collisions of random strength, due to the chaotic water molecules, in any time interval much smaller than the observation time. Simultaneously, in 1905 Karl Pearson raised interest in the corresponding discrete-time problem, namely the drunkard's walk, which inherited the name random walk from the way it was originally formulated by Pearson himself. More specifically, given n consecutive vectors denoting position displacements of equal modulus and random orientation (assuming that each direction is equally probable), one can inquire about the distance of a man, following the trajectory induced by the sequence of vectors, from the starting position of the walk after n steps. Plenty of exact analytical results have been obtained thereafter for spatially homogeneous models. Nevertheless, as soon as one works with more or less significant perturbations of this substantial family of processes, classical methods fail and the available literature is much less comprehensive. Despite this doubtelss demanding perspective, inhomogeneity naturally earned an ever-growing interest over the years due to its far more realistic applications. In this thesis, we tried to shed some light on certain statistical, probabilistic or dynamical aspects of three different spatially non-homogeneous systems. The first chapter examines a research project in probability theory in which I took part since June 2020, which turned out to be the most challenging work of these last three years. It was initially conceived with the aim to characterize the first-passage events for a particular class of stochastic processes with correlated increments, the so called random walks in a one-dimensional Lévy random medium, where correlations are created by performing a classical random walk on a fixed random array of targets instead of a regular lattice. The lacking of the independence hypothesis has generated implications harder to handle than we ever imagined. Anyway, in the end we still managed to get a statement more general than expected, even though with some technical mathematical requirements in the most problematic subcases. The second chapter refers to my original Ph.D. project and fits within the analysis of Lamperti-type discrete-time stochastic processes, namely nearest-neighbour random walks with an asymptotically-vanishing position-dependent drift, whose special interest lies in the possibility of deeply understanding a sort of phase transition in the asymptotic behaviour of classical random walk models. We started by considering the centrally-biased Gillis model, which is a historical benchmark in this context, and extended the study beyond the well-known recurrence properties. After characterizing the different regimes, we shifted our attention to generalized versions of this random walk and compared all our findings to problems already addressed in the literature. The third and final chapter, instead, concerns a collaboration with another Ph.D. student born during a summer school in statistical physics at the end of the second year (July 2021). We made one-dimensional classical random walks with independent and identically distributed jumps non-homogeneous by introducing periodically distributed holes of finite size in the phase space, that is the real line. We looked at the statistical properties of the trajectories, with an emphasis on the survival probability and the characteristic time of the escape event, which subsequently allowed us to derive the transport properties of the particles conditioned to survive. In particular, our results unveiled an intriguing connection with deterministic open dynamical systems, for instance strongly chaotic planar billiards.