On exact solutions for non-homogeneous diffusions

Among all stochastic processes, Brownian motion plays a crucial role mainly for being one of the simplest models of diffusions. One of hallmark features of this process, is the linear growth of the mean squared displacement ⟨x^2(t)⟩ ∝ t with time. Over the years, an increasing number of physical systems was reported to violate this linear dependence. Therefore, the interest in mathematical generalizations of the Brownian motion has significantly grown over the years. From a mathematical perspective, Brownian motion is a stochastic process characterized by independent, stationary and normally distributed increments. The main idea of this dissertation is to explore some generalizations of Brownian motion by removing either one or more of the aforementioned three hypothesis. The thesis is articulated into two main areas: the first one investigates processes characterized by discontinuous increments, notably processes with partial stochastic resetting (PSR) or Poissonian-shot noise (PSN); the second part studies processes characterized by non-stationary increments, with a special focus on non-autonomous diffusions. The results obtained in the former part, for what concerns PSR, include: a complete study of isotropic d-dimensional Lévy flights with PSR, notably the derivation of the propagators of the processes, the moments of the time-dependent space distribution, the large deviation scaling of the propagators with the related dynamical phase transition, and the behavior of the propagators around the typical value. For what concerns PSN, instead, the results of the dissertation include: the study of Brownian motion with PSN with the calculation of its stationary density and its moments, a possible application of the model to an experiment with bacteria, an efficient clear-cut inference scheme relating the microscopic parameters of the model to experimentally measurable quantities. The second part of the dissertation deals with the more general problem of solving stochastic differential equations (SDEs) and related observables with arbitrary space and time-dependent drift and volatility parameters. The first main result of this part is the development of a general framework to solve SDEs and their related first passage time densities. The developed technique is then applied to a couple of models, one used in experiments with optical tweezers and the other in population dynamics. The results of this part include several exact analytical formulae of propagators, autocorrelations and first-passage time densities for non-autonomous processes.