Probability Theory and Related Fields

, Volume 129, Issue 4, pp 469–494

Exact L2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems


    • Department of Mathematics and MechanicsSt.Petersburg State University
  • Ya.Yu. Nikitin
    • Department of Mathematics and MechanicsSt.Petersburg State University

DOI: 10.1007/s00440-004-0337-z

Cite this article as:
Nazarov, A. & Nikitin, Y. Probab. Theory Relat. Fields (2004) 129: 469. doi:10.1007/s00440-004-0337-z


We find the exact small deviation asymptotics for the L2-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.

Key words or phrases:Integrated Wiener processIntegrated OrnsteinUhlenbeck processSmall deviationsBoundary value problemGreen function; Spectral asymptotics

Copyright information

© Springer-Verlag Berlin Heidelberg 2004