  Suppose that $\mathsf{S}$ is a closed set of the unit sphere
$\mathbb{S}^{d-1} = \{x\in \mathbb{R}^d: |x| =1\}$ in dimension $d\geq2$, which
has positive surface measure. We construct the law of absorption of an
isotropic stable L\'evy process in dimension $d\geq2$ conditioned to approach
$\mathsf{S}$ continuously, allowing for the interior and exterior of
$\mathbb{S}^{d-1}$ to be visited infinitely often. Additionally, we show that
this process is in duality with the underlying stable L\'evy process. We can
replicate the aforementioned results by similar ones in the setting that
$\mathsf{S}$ is replaced by $\mathsf{D}$, a closed bounded subset of the
hyperplane $\{x\in\mathbb{R}^d : (x, v) = 0\}$ with positive surface measure,
where $v$ is the unit orthogonal vector and where $(\cdot,\cdot )$ is the usual
Euclidean inner product. Our results complement similar results of the authors
Kyprianou, Palau and Saizmaa (2020) in which the stable process was further
constrained to attract to and repel from $\mathsf{S}$ from either the exterior
or the interior of the unit sphere.
