We consider a class of spatially non-homogeneous random walks in multidimensional Euclidean space with zero drift, which in any dimension (two or higher) can be recurrent or transient depending on the details of the walk. These walks satisfy an invariance principle, and have as their scaling limits a class of martingale diffusions, with law determined uniquely by an SDE with discontinuous coefficients at the origin. Furthermore, pathwise uniqueness of this SDE may fail. The radial coordinate of the diffusion is a Bessel process of dimension greater than 1. Unique characterization of the law of the diffusion, which must start at the origin, is natural via excursions built around the Bessel process; each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining appropriately the Riemannian metric $g$ on the sphere $S^{d-1}$ allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Browninan motion with drift on the Riemannian manifold $(S^{d-1},g)$. In particular, this provides a multidimensional generalisation of the Pitman-Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace-Beltrami operator and the divergence under the metric $g$.

This is joint work with Nicholas Georgiou and Andrew Wade.

I will discuss the motion of hybrid zones, in the context of the Spatial Lambda-Fleming-Viot process. Hybrid zones are thin interfaces that form between populations which are genetically distinct, but still able to interbreed. The work relies on a new connection between branching Brownian motion, the Allen-Cahn equation and mean curvature flow.

The many flows asymptotic for queueing systems was introduced by Alan Weiss in 1986 is a natural asymptotic for large systems. In this talk we introduce a novel scaling framework for this asymptotic and examine how various results relate to each other. To introduce the framework we consider a simple Markovian example before exploring some of the sample path large deviations principles obtainable. We then utilize these results to gain a better insight into the multiplexing gain obtained from pooling resources. Finally we introduce some open problems which are thrown up by this new framework.

For infections which can become endemic in a population, a random variable of particular interest is the time until extinction of infection. Starting from a population in which infection is endemic, which is to say that the population is initiated in a quasi-stationary state, then the time to extinction of infection is exponentially distributed, so that it is sufficient to approximate its mean. An approach based upon ideas from Hamiltonian mechanics has recently received a lot of attention in the theoretical physics literature, but not so much in the applied probability or mathematical biology literature. I will review this technique from an applied probabilist’s perspective, giving examples of applications to a number of well-known stochastic infection models.