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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 $d\geq2$, whichhas positive surface measure. We construct the law of absorption of anisotropic 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 thatthis process is in duality with the underlying stable L\'evy process. We canreplicate the aforementioned results by similar ones in the setting that$\mathsf{S}$ is replaced by $\mathsf{D}$, a closed bounded subset of thehyperplane $\{x\in\mathbb{R}^d : (x, v) = 0\}$ with positive surface measure,where $v$ is the unit orthogonal vector and where $(\cdot,\cdot )$ is the usualEuclidean inner product. Our results complement similar results of the authorsKyprianou, Palau and Saizmaa (2020) in which the stable process was furtherconstrained to attract to and repel from $\mathsf{S}$ from either the exterioror the interior of the unit sphere.

Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable L\'evy processes