Abstract

Diffusion is a very common model for motion in physics, chemistry, biology and in the social sciences. Typically, populations of particles or molecules, or individuals tend to ‘equilibrate’ by moving into less populated areas. The macroscopic phenomenon of linear diffusion is generated by the microscopic phenomenon of random Brownian motion (the infinitesimal limit of short uncorrelated steps taken at small time intervals). The latter is a commonly used model to represent uncertainty in particle motion. However, much more complicated processes occur in many important applications, such as diffusion of a non-infinitesimal non-local nature, where the motion at a given point in space is influenced by events at many different scales. Such processes arise in many different contexts, for example in continuum mechanics when surface diffusion is influenced by spatial considerations (semipermeable membranes, the quasigeostrophic equation for atmospheric and oceanic flows) and in problems involving phase transitions. On a probabilistic level non-infinitesimal diffusions are often described by Levy-type jump processes (a significant generalisation of Brownian motion).

We shall discuss some important nonlinear partial differential problems involving such diffusion processes.