## Abstract

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 language | English |
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Title of host publication | A Lifetime of Excursions Through Random Walks and Lévy Processes |

Editors | L. Chaumont , A. E. Kyprianou |

Place of Publication | Cham, Switzerland |

Publisher | Birkhauser |

Pages | 283-313 |

Number of pages | 31 |

ISBN (Electronic) | 9783030833091 |

ISBN (Print) | 9783030833084 |

DOIs | |

Publication status | E-pub ahead of print - 30 Jul 2021 |

### Publication series

Name | Progress in Probability |
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Volume | 78 |

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.

## Keywords

- Duality
- Stable process
- Time reversal

## ASJC Scopus subject areas

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