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