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Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems
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  • Published: 25 May 2004

Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

  • A.I. Nazarov1 &
  • Ya.Yu. Nikitin1 

Probability Theory and Related Fields volume 129, pages 469–494 (2004)Cite this article

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Abstract.

We find the exact small deviation asymptotics for the L 2 -norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, St.Petersburg State University, 198504, Russia

    A.I. Nazarov & Ya.Yu. Nikitin

Authors
  1. A.I. Nazarov
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  2. Ya.Yu. Nikitin
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Correspondence to A.I. Nazarov.

Additional information

Partially supported by grants of RFBR 01-01-00245 and 02-01-01099.

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Nazarov, A., Nikitin, Y. Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems. Probab. Theory Relat. Fields 129, 469–494 (2004). https://doi.org/10.1007/s00440-004-0337-z

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  • Received: 05 April 2003

  • Revised: 06 January 2004

  • Published: 25 May 2004

  • Issue Date: August 2004

  • DOI: https://doi.org/10.1007/s00440-004-0337-z

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  • Key words or phrases:
  • Integrated Wiener process
  • Integrated Ornstein
  • Uhlenbeck process
  • Small deviations
  • Boundary value problem
  • Green function; Spectral asymptotics
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