Recent Developments in Anomalous Diffusions

Abstract

In this paper we review briefly some of the results in the area of anomalous diffusions which are related to the anomalous aggregation phenomenon. Loosely, speaking this phenomenon occurs when a particle moves in a milieu with obstacles which alter so much its otherwise Markovian or diffusive motion that instead of free movement throughout the environment the particle tends to spend a predominant proportion of time in the vicinity of the strongest traps. This type of behaviour is observed in systems such as human cells, polluted rivers, motion in porous media, etc. We also offer some historic account on the appearance of anomalous diffusion in science and the main contributions in this contemporary area of mathematics. We also present our published results with Bruno Toaldo (Turin, Italy) on the topic of anomalous aggregation in this very active field of research. We also discuss some open problems and future directions for research which present a formidable technical challenge. 2020 Mathematics Subject Classification: 60K15, 60J65, 60J25, 60G51. 2020 Mathematics Subject Classification: 60K15, 60J65, 60J25, 60G51.