Diffusione e reazione: dal moto Browniano alla diffusione delle epidemie

Diffusion and transport processes constitute a very important field of applied mathematics. They are useful in many different problems ranging from the diffusion of pollutants in the atmosphere and the sea, to the spreading of epidemics. Aside from their practical relevance, such processes have been very important in the history of physics and mathematics. We can recall Einstein's study of Brownian motion which was fundamental to give a definitive experimental evidence of the existence of atoms. Moreover diffusion processes have been the starting point for the building of the mathematical theory of stochastic processes (starting from the work of Langevin). Similarly the study of reaction and diffusion phenomena, starting from the seminal contribution of two 20th-century science giants (Ronald A. Fisher and Andrej N. Kolmogorov) has led to interesting developments both for applications and for the fruitful connections between stochastic processes and partial differential equation. In this article we discuss some general results developed in these areas, including few modern topics, as transport and reaction/diffusion on discrete structures (graphs). Such a theme has a great relevance, e.g. for the dissemination of information through the internet or the spreading of epidemics through the air transport network.