  We prove an invariance principle for a class of zero-drift spatially
non-homogeneous random walks in $\mathbb{R}^d$, which may be recurrent in any
dimension. The limit $\mathcal{X}$ is an elliptic martingale diffusion, which
may be point-recurrent at the origin for any $d\geq2$. To characterise
$\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit
sphere in $\mathbb{R}^d$ and use it to express a related spherical diffusion as
a Brownian motion with drift. This representation allows us to establish the
skew-product decomposition of the excursions of $\mathcal{X}$ and thus develop
the excursion theory of $\mathcal{X}$ without appealing to the strong Markov
property. This leads to the uniqueness in law of the stochastic differential
equation for $\mathcal{X}$ in $\mathbb{R}^d$, whose coefficients are
discontinuous at the origin. Using the Riemannian metric we can also detect
whether the angular component of the excursions of $\mathcal{X}$ is
time-reversible. If so, the excursions of $\mathcal{X}$ in $\mathbb{R}^d$
generalise the classical Pitman-Yor splitting-at-the-maximum property of Bessel
excursions.
