Outline:

It is a natural question to ask how the behaviour of a stochastic process depends on the geometry of the space in which it takes values. When the underlying space is itself random (e.g. a percolation cluster), sometimes unusual, or even surprising, phenomena can occur. For instance, inhomogeneities in the environment can lead to acceleration or
trapping phenomena that are not seen for deterministic spaces. I can offer various projects exploring this area, and am happy to discuss further details with interested candidates. Those wishing to study such problems with me should have a strong background in probability, including stochastic processes.