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



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 process Integrated Ornstein Uhlenbeck process Small deviations Boundary value problem Green function; Spectral asymptotics 


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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsSt.Petersburg State UniversityRussia

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