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A Lifetime of Excursions Through Random Walks and Lévy Processes Read chapter Read first chapter

2021 | OriginalPaper | Chapter

Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes

Authors : Mateusz Kwaśnicki, Andreas E. Kyprianou, Sandra Palau, Tsogzolmaa Saizmaa

Published in: A Lifetime of Excursions Through Random Walks and Lévy Processes

Publisher: Springer International Publishing

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Abstract

Suppose that S is a closed set of the unit sphere \(\mathbb {S}^{d-1} = \{x\in \mathbb {R}^d: |x| =1\}\) in dimension d ≥ 2, which has positive surface measure. We construct the law of absorption of an isotropic stable Lévy process in dimension d ≥ 2 conditioned to approach 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 unconditioned stable Lévy process. We can replicate the aforementioned results by similar ones in the setting that S is replaced by 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 (⋅, ⋅) is the usual Euclidean inner product. Our results complement similar results of the authors [17] in which the stable process was further constrained to attract to and repel from S from either the exterior or the interior of the unit sphere.

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previous chapter The Doob–McKean Identity for Stable Lévy Processes
next chapter Angular Asymptotics for Random Walks
Appendix
Available only for authorised users
Footnotes
1
We will distinguish integrals with respect to one-dimensional Lebesgue measure as taking the form \(\int \cdot \, \mathrm {d} x\), where as higher dimensional integrals will always indicate the dimension, for example \(\int \cdot \, \ell _d(\mathrm {d} x)\).
 
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Metadata
Title
Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes
Authors
Mateusz Kwaśnicki
Andreas E. Kyprianou
Sandra Palau
Tsogzolmaa Saizmaa
Copyright Year
2021
Book
A Lifetime of Excursions Through Random Walks and Lévy Processes

Print ISBN: 978-3-030-83308-4

Electronic ISBN: 978-3-030-83309-1

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-83309-1_16