Parabolic Anderson Model on sparse random graphs


The Parabolic Anderson Model is a heat equation with a random potential. It describes the diffusion of heat or particles in the presence of sources and sinks. So far the PAM has been studied mostly on lattices and on Euclidean space, and mostly for random potentials that have a very short correlation length. The main interest is in the long-time asymptotics of the total  mass of the solution to the PAM equation and the question where the bulk of the total mass is mainly concentrated, i.e., where is it located and how large is it as a function of time?

A key quantity of interest is the so-called Lyapunov exponent, the logarithmic growth rate of the total mass. Here one may distinguish between the annealed setting (i.e., taking the expectation with respect to the random potential) and the quenched setting (i.e., almost surely with respect to the random potential). The main contribution to the total mass typically comes from a single `intermittent island’ in space that is far away from the location of the initial mass. On this island the potential typically has a certain characteristic shape. 

The above picture has been verified in a variety of different settings. The goal of the project is to study the PAM on sparse random graphs, which have the property that they are `locally tree-like’, i.e., on a small scale look like a random tree. The target is to understand what the intermittent island looks like and how its size and shape can be characterised.

The project relies on probability theory, functional analysis and spectral theory. Mathematical techniques are rooted in large deviation theory and variational calculus. 

Supervisors Frank den Hollander (UL), Wolfgang König, Renato Dos Santos
PhD Student Daoyi Wang
Location Universiteit Lieden (UL)