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

doi:10.1007/978-3-030-83309-1_16

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 proceeding › Chapter or section

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
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	Birkhäuser
Pages	283-313
Number of pages	31
ISBN (Electronic)	9783030833091
ISBN (Print)	9783030833084
DOIs	https://doi.org/10.1007/978-3-030-83309-1_16
Publication status	E-pub ahead of print - 30 Jul 2021

Publication series

Name	Progress in Probability
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)

Access to Document

10.1007/978-3-030-83309-1_16

Cite this

Kwaśnicki, M., Kyprianou, A. E., Palau, S., & Saizmaa, T. (2021). Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes. In L. Chaumont , & A. E. Kyprianou (Eds.), A Lifetime of Excursions Through Random Walks and Lévy Processes (pp. 283-313). (Progress in Probability; Vol. 78). Birkhäuser. Advance online publication. https://doi.org/10.1007/978-3-030-83309-1_16

Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes. / Kwaśnicki, Mateusz; Kyprianou, Andreas E.; Palau, Sandra et al.
A Lifetime of Excursions Through Random Walks and Lévy Processes. ed. / L. Chaumont ; A. E. Kyprianou. Cham, Switzerland: Birkhäuser, 2021. p. 283-313 (Progress in Probability; Vol. 78).

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

Kwaśnicki, M, Kyprianou, AE, Palau, S & Saizmaa, T 2021, Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes. in L Chaumont & AE Kyprianou (eds), A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol. 78, Birkhäuser, Cham, Switzerland, pp. 283-313. https://doi.org/10.1007/978-3-030-83309-1_16

Kwaśnicki M, Kyprianou AE, Palau S, Saizmaa T. Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes. In Chaumont L, Kyprianou AE, editors, A Lifetime of Excursions Through Random Walks and Lévy Processes. Cham, Switzerland: Birkhäuser. 2021. p. 283-313. (Progress in Probability). Epub 2021 Jul 30. doi: 10.1007/978-3-030-83309-1_16

Kwaśnicki, Mateusz ; Kyprianou, Andreas E. ; Palau, Sandra et al. / Oscillatory Attraction and Repulsion from a Subset of the Unit Sphere or Hyperplane for Isotropic Stable Lévy Processes. A Lifetime of Excursions Through Random Walks and Lévy Processes. editor / L. Chaumont ; A. E. Kyprianou. Cham, Switzerland : Birkhäuser, 2021. pp. 283-313 (Progress in Probability).

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

Abstract

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Bibliographical note

Keywords

ASJC Scopus subject areas

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