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Persistence Probabilities and Exponents

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Lévy Matters V

Part of the book series: Lecture Notes in Mathematics ((LEVY,volume 2149))

Abstract

This article deals with the asymptotic behavior as \(t \rightarrow +\infty \) of the survival function \(\mathbb{P}[T > t]\), where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type \(\mathbb{P}[T > t] = t^{-\theta +o(1)}\) for a known or unknown positive parameter \(\theta\) which is called the persistence exponent. The problem is well understood for random walks or Lévy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

AMS Subject Classification 2000: 60F99, 60G10, 60G15, 60G18, 60G50, 60G52, 60K35, 60K40

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Acknowledgements

This work was supported by the DFG Emmy Noether research program and the grant ANR-09-BLAN-0084-01 Autosimilaires. The authors would like to thank R.A. Doney, S.N. Majumdar, A.A. Novikov and the referees for their useful comments.

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Authors and Affiliations

  1. Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Frank Aurzada

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

    Thomas Simon

  3. Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris Sud, Bâtiment 100, F-91405, Orsay Cedex, France

    Thomas Simon

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Aurzada, F., Simon, T. (2015). Persistence Probabilities and Exponents. In: Lévy Matters V. Lecture Notes in Mathematics(), vol 2149. Springer, Cham. https://doi.org/10.1007/978-3-319-23138-9_3

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