Abstract:

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. The radial coordinate of the diffusion is a Bessel process of dimension greater than 1 (this component of the invariance principle is related to a theorem of Lamperti). Unique characterization in 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 Remannian metric on the sphere allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) of the process. This is joint work with Nicholas Georgiou and Andrew Wade.