Homogenization of random walks: degenerate environments and long-range jumps

Consider a lattice consisting of vertices and edges. To each edge we assign a randomly chosen positive number called conductance. Now consider a random walk (or particle) moving along the vertices of the lattice in such a way that the probability to jump from one vertex to one of its neighbours is proportional to the conductance on the connecting edge. This model of a reversible random walk in a random environment is known in the literature as the random conductance model.

Random walks in random environments have been at the centre of the interest in probability theory for several decades. One motivation originates from applications in physics, material science or biology, as for instance the study of transport processes through porous media or in composite materials. A common characteristics of such heterogeneous media is the presence of strong spatial inhomogeneities on microscopic scales. Since the microscopic structure can often be characterised only statistically, such transport processes in a heterogeneous medium are naturally modelled by random walks in random environment. When studying such random walks on macroscopic length and time-scales, which are much larger compared to the microscopic heterogeneities, one typically observes that the random irregular microstructures are averaged out and homogenisation effects arise, so that the effective macroscopic behaviour can be described by a much simpler stochastic process in a homogeneous environment.

Mathematically, it is now of interest under which conditions on the random medium such homogenisation effects occur. This can be formulated in terms of scaling limit results for the random walk. In this project we aim to establish such scaling limits for random walks under random conductances with long range jumps, i.e. the random walk is not only allowed to jump to one of its neighbours but also to other vertices further away. As the underlying set of vertices we consider (1) the Euclidean lattice and (2) the realisation of a point process in the Euclidean space.

We also aim to study the associated partial differential differential equations (PDE) involving non-local discrete operators describing the transition probabilities of such random walks. In fact, random conductance models are of interest to PDE analysts as well as probabilists because the tools that are used to study them borrow techniques from both fields. There are also strong links to mathematical physics.