Spectral theory for symmetric one-dimensional Lévy processes killed upon hitting the origin

Mateusz Kwaśnicki (Polish Academy of Sciences and Wrocław University of Technology)

Abstract


Spectral theory for transition operators of one-dimensional symmetric Lévy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is proved. As an application, and the primary motivation, integral fomulae for the transition density and the distribution of the hitting time of the origin are given in terms of the eigenfunctions.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-29

Publication Date: October 1, 2012

DOI: 10.1214/EJP.v17-2013

References

  • Bañuelos, Rodrigo; Kulczycki, Tadeusz. The Cauchy process and the Steklov problem. J. Funct. Anal. 211 (2004), no. 2, 355--423. MR2056835
  • Bañuelos, Rodrigo; Kulczycki, Tadeusz. Trace estimates for stable processes. Probab. Theory Related Fields 142 (2008), no. 3-4, 313--338. MR2438694
  • Bertoin, Jean. Lévy Processes. Cambridge University Press, Melbourne-New York, 1998.
  • Blumenthal, R. M.; Getoor, R. K.; Ray, D. B. On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 1961 540--554. MR0126885
  • Cordero, Fernando. Sur la théorie des excursions pour des processus de Lévy symétriques stables d'indice α in ]1, 2], et quelques applications. PhD thesis, Université Paris 6, 2010.
  • Doney, R. A. Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. (2) 44 (1991), no. 3, 566--576. MR1149016
  • Dynkin, E. B. Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg 1965 Vol. I: xii+365 pp.; Vol. II: viii+274 pp. MR0193671
  • Frank, Rupert L.; Geisinger, Leander. Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain. Mathematical results in quantum physics, 138--147, World Sci. Publ., Hackensack, NJ, 2011. MR2885166
  • Kesten, Harry. A convolution equation and hitting probabilities of single points for processes with stationary independent increments. Bull. Amer. Math. Soc. 75 1969 573--578. MR0251797
  • Kaleta, Kamil; Kwasnicki, Mateusz; Malecki, Jacek. One-dimensional quasi-relativistic particle in the box. Preprint 2011, arXiv:1110.5887v1.
  • Kulczycki, Tadeusz; Kwaśnicki, Mateusz; Małecki, Jacek; Stos, Andrzej. Spectral properties of the Cauchy process on half-line and interval. Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 589--622. MR2679702
  • Kwaśnicki, Mateusz. Spectral analysis of subordinate Brownian motions on the half-line. Studia Math. 206 (2011), no. 3, 211--271. MR2860308
  • Kwasnicki, Mateusz; Malecki, Jacek; Ryznar, Michal. First passage times for subordinate Brownian motions. Preprint 2011, arXiv:1110.0401v1.
  • Kwaśnicki, Mateusz. Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262 (2012), no. 5, 2379--2402. MR2876409
  • Port, Sidney C. Hitting times and potentials for recurrent stable processes. J. Analyse Math. 20 1967 371--395. MR0217877
  • Peskir, Goran. The law of the hitting times to points by a stable Lévy process with no negative jumps. Electron. Commun. Probab. 13 (2008), 653--659. MR2466193
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Cambridge Univ. Press, Cambridge, 1999.
  • Schilling, René L.; Song, Renming; Vondraček, Zoran. Bernstein functions. Theory and applications. de Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, 2010. xii+313 pp. ISBN: 978-3-11-021530-4 MR2598208
  • Vladimirov, V. S. Methods of the theory of generalized functions. Analytical Methods and Special Functions, 6. Taylor & Francis, London, 2002. xiv+311 pp. ISBN: 0-415-27356-0 MR2012831
  • Yano, Kouji. Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. Potential Anal. 32 (2010), no. 4, 305--341. MR2603019
  • Yano, Kouji; Yano, Yuko; Yor, Marc. On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes. In: Séminaire de Probabilités XLII, Lecture Notes in Math. 1979, Springer, 2009.
  • Yano, Kouji; Yano, Yuko; Yor, Marc. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan 61 (2009), no. 3, 757--798. MR2552915
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.