Taking account of recent developments in the representation of $d$-dimensional isotropic stable Lévy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that $\mathsf{S}$ is a region of the unit sphere ${\mathbb{S}}^{d-1}=\{x\in {\mathbb{R}}^d:\left|x\right|=1\}$. We construct the aforesaid stable Lévy process conditioned to approach $\mathsf{S}$ continuously from either inside or outside of the sphere. Additionally, we show that these processes are in duality with the stable process conditioned to remain inside the sphere and absorb continuously at the origin and to remain outside of the sphere, respectively. Our results extend the recent contributions of Döring and Weissman (2020), where similar conditioning is considered, albeit in one dimension as well as providing analogues of the same and very classical results for Brownian motion, (Doob, 1957). As in Döring and Weissman (2020), we appeal to recent fluctuation identities related to the deep factorisation of stable processes, cf. (Kyprianou, 2016; Kyprianou et al., 2020; Kyprianou et al., 2017).

考虑到以下方面的最新发展 $d$作为自相似马尔可夫过程的三维各向同性稳定Lévy过程，我们考虑了许多限制其路径的新方法。假设$\mathsf{小号}$ 是单位球体的区域 ${\mathbb{小号}}^{d-\mathrm{1个}}=\{X\in {\mathbb{[R}}^d：\left|X\right|=\mathrm{1个}\}$。我们构建了满足以下条件的上述稳定的Lévy过程$\mathsf{小号}$从球体内部或外部连续不断地传播。此外，我们证明了这些过程是对偶的，稳定过程的条件是分别保留在球体内部并在原点连续吸收并保持在球体外部。我们的研究结果扩展了Döring和Weissman（2020）的最新研究成果，该研究考虑了类似的条件，尽管是一维的，并且为布朗运动提供了相同且非常经典的结果的类似物（Doob，1957）。就像在Döring和Weissman（2020）中一样，我们呼吁与稳定过程的深度分解相关的近期波动特性，请参见。（Kyprianou，2016; Kyprianou等，2020; Kyprianou等，2017）。