论文标题
各向同性稳定过程的单位球或超平面子集的振荡吸引力和排斥
Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes
论文作者
论文摘要
假设$ \ mathsf {s} $是一组封闭的单位球体$ \ mathbb {s}^{d-1} = \ {x \ in \ mathbb {r}^d：| x | | x | | = 1 \} $ in dimension $ d \ geq2 $，具有正面度量。我们构建了在尺寸中吸收各向同性稳定的lévy过程的定律，以连续接近$ \ mathsf {s} $接近$ \ mathsf {s} $，允许内部和外部的内部和外部经常被无限地访问。此外，我们表明此过程与基本稳定的lévy过程是双重性。 我们可以在$ \ mathsf {s} $被$ \ mathsf {d} $替换的情况下，通过类似结果复制上述结果，这是超平面$ \ {x \ in \ mathbb {r}^r}^d：（x，x，x，x，x，x，x，x，x，x，v）$ v y的封闭界面子集的封闭式子集，其中$（\ cdot，\ cdot）$是通常的欧几里得内部产品。 我们的结果补充了作者Kyprianou，Palau和Saizmaa（2020）的相似结果，其中稳定过程受到进一步限制，以吸引并从单位球体的外部或内部从$ \ mathsf {s} $中击退。
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 Lévy 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 Lévy 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.
