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

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

Research output: Chapter or section in a book/report/conference proceedingChapter or section


Suppose that S is a closed set of the unit sphere Sd−1 = {x ∈ Rd : |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 Sd−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 ∈ Rd : (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.

Original languageEnglish
Title of host publicationA Lifetime of Excursions Through Random Walks and Lévy Processes
EditorsL. Chaumont , A. E. Kyprianou
Place of PublicationCham, Switzerland
Number of pages31
ISBN (Electronic)9783030833091
ISBN (Print)9783030833084
Publication statusE-pub ahead of print - 30 Jul 2021

Publication series

NameProgress in Probability
ISSN (Print)1050-6977
ISSN (Electronic)2297-0428

Bibliographical note

Funding Information:
Acknowledgments TS acknowledges support from a Schlumberger Faculty of the Future award. SP acknowledges support from the Royal Society as a Newton International Fellow Alumnus (AL201023) and UNAM-DGAPA-PAPIIT grant no. IA103220.

Publisher Copyright:
© 2021, Springer Nature Switzerland AG.


  • Duality
  • Stable process
  • Time reversal

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics
  • Mathematical Physics
  • Mathematics (miscellaneous)


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