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On the Harmonic Measure of Stable Processes

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Séminaire de Probabilités XLVIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2168))

Abstract

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real Lévy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the hitting probabilities of points and describe the non-negative harmonic functions for the stable process killed outside a finite interval.

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Notes

  1. 1.

    Among of course many others. This one is a simple consequence of the two-dimensional structure of the space of solutions to the hypergeometric equation. Notice that it can also be obtained by Mellin-Barnes inversion. See the end of the article Calculs asymptotiques in Encyclopedia Universalis.

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Acknowledgements

Nous savons gré à Jean Jacod d’un chat instructif sur la distance de Skorokhod. C. P. a bénéficié du support de la Chaire Marchés en Mutation, Fédération Bancaire Française. Travail dédié à l’association Laplace-Gauss.

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Authors and Affiliations

  1. Laboratoire de mathématiques et modélisation d’Evry, Université d’Evry-Val d’Essonne, F-91037, Evry Cedex, France

    Christophe Profeta

  2. Laboratoire Paul Painlevé, Université Lille 1, F-59655, Villeneuve d’Ascq Cedex, France

    Thomas Simon

  3. Laboratoire de physique théorique et modèles statistiques, Université Paris-Sud, F-91405, Orsay Cedex, France

    Thomas Simon

Authors
  1. Christophe Profeta
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  2. Thomas Simon
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Correspondence to Christophe Profeta .

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Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles , France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandoeuvre-les-Nancy CEDEX, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

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Profeta, C., Simon, T. (2016). On the Harmonic Measure of Stable Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVIII. Lecture Notes in Mathematics(), vol 2168. Springer, Cham. https://doi.org/10.1007/978-3-319-44465-9_12

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