Abstract: In this talk we describe an invariance principle for a class of non-homogeneous martingale random walks in $\RR^d$ that can be recurrent or transient for any dimension $d$. The scaling limit, which we construct, is a martingale diffusions with law determined uniquely by an SDE with discontinuous coefficients at the origin whose pathwise uniqueness may fail. The radial component of the diffusion is a Bessel process of dimension greater than 1. We characterize the law of the diffusion, which must start at the origin, via its 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 a Riemannian metric $g$ on the sphere $S^{d−1}$, different from the one induced by the ambient Euclidean space, 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.
